A1

A Review of Basic Algebra

CAREERS AND MATHEMATICS: PharmacistPharmacists distribute prescription drugs to individuals. They also advise patients, physicians, and other health-care workers on the selection, dosages, interactions, and side effects of medi-cations. They also monitor patients to ensure that they are using their medica-tions safely and effectively. Some phar-macists specialize in oncology, nuclear pharmacy, geriatric pharmacy, or psy-chiatric pharmacy.

Education and Mathematics Required• Pharmacists are required to possess a Pharm.D. degree from an accredited col-

lege or school of pharmacy. This degree generally takes four years to complete. To be admitted to a Pharm.D. program, at least two years of college must be completed, which includes courses in the natural sciences, mathematics, human-ities, and the social sciences. A series of examinations must also be passed to obtain a license to practice pharmacy.

• College Algebra, Trigonometry, Statistics, and Calculus I are courses required for admission to a Pharm.D. program.

How Pharmacists Use Math and Who Employs Them• Pharmacists use math throughout their work to calculate dosages of various

drugs. These dosages are based on weight and whether the medication is given in pill form, by infusion, or intravenously.

• Most pharmacists work in a community setting, such as a retail drugstore, or in a healthcare facility, such as a hospital.

Career Outlook and Salary• Employment of pharmacists is expected to grow by 17 percent between 2008

and 2018, which is faster than the average for all occupations.• Median annual wages of wage and salary pharmacists is approximately

$106,410.

For more information see: www.bls.gov/oco

A A.1 Sets of Real Numbers

A.2 Integer Exponents and Notation

A.3 Rational Exponents and Radicals

A.4 Polynomials

A.5 Factoring Polynomials

A.6 Rational Expressions

Appendix Review

Appendix Test

In this appendix, we review many concepts and skills learned in previous algebra courses.

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A2 AppendixAAReviewofBasicAlgebra

A.1 Sets of Real NumbersInthissection,wewilllearnto

1.Identifysetsofrealnumbers.2.Identifypropertiesofrealnumbers.3.Graphsubsetsofrealnumbersonthenumberline.4.Graphintervalsonthenumberline.5.Defineabsolutevalue.6.Finddistancesonthenumberline.

Sudoku, a game that involves number placement, is very popular. The objective is to fill a 9 by 9 grid so that each column, each row and each of the 3 by 3 blocks contains the numbers from 1 to 9. A partially completed Sudoku grid is shown in the margin.

To solve Sudoku puzzles, logic and the set of numbers, 51, 2, 3, 4, 5, 6, 7, 8, 96are used. Sets of numbers are important in mathematics, and we begin our study of algebra with this topic.

A set is a collection of objects, such as a set of dishes or a set of golf clubs. The set of vowels in the English language can be denoted as 5a, e, i, o, u6, where the braces 5 6 are read as “the set of.”

If every member of one set B is also a member of another set A, we say that B is a subset of A. We can denote this by writing B ( A, where the symbol ( is read as “is a subset of.” (See Figure A.1 below.) If set B equals set A, we can write B 8 A.

If A and B are two sets, we can form a new set consisting of all members that are in set A or set B or both. This set is called the union of A and B. We can denote this set by writing, A c B where the symbol c is read as “union.” (See Figure A.1 below.)

We can also form the set consisting of all members that are in both set A and set B. This set is called the intersection of A and B. We can denote this set by writing A d B, where the symbol d is read as “intersection.” (See Figure A.1 below.)

A

B

B � A

A B

A � B

A B

A � B

FiGUre A-1

eXaMPLe 1 Understanding subsets and Finding the Union and intersection of two sets

Let A 5 5a, e, i6, B 5 5c, d, e6, and V 5 5a, e, i, o, u6.a. Is A ( V? b. Find A c B. c. Find A d B.

sOLUtiOn a. Since each member of set A is also a member of set V, A ( V .

b. The union of set A and set B contains the members of set A, set B, or both. Thus, A c B 5 5a, c, d, e, i6.

c. The intersection of set A and set B contains the members that are in both set A and set B. Thus, A d B 5 5e6.

self Check 1 a. Is B ( V? b. Find B c V c. Find A d V

now try exercise 33.

5 3 76

8 6 34 8 3 17 2 6

9

64 1

8

SUDOKU

7 99 5

2 8

8 691 5

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SectionA.1SetsofRealNumbersA3

rational numbers

Basic sets of numbers

1. identify sets of real numbersThere are several sets of numbers that we use in everyday life.

NaturalnumbersThe numbers that we use for counting: 51, 2, 3, 4, 5, 6, % 6WholenumbersThe set of natural numbers including 0: 50, 1, 2, 3, 4, 5, 6, % 6IntegersThe set of whole numbers and their negatives:5 % , 25, 24, 23, 22, 21, 0, 1, 2, 3, 4, 5, % 6

In the definitions above, each group of three dots (called an ellipsis) indicates that the numbers continue forever in the indicated direction.

Two important subsets of the natural numbers are the prime and composite numbers. A primenumber is a natural number greater than 1 that is divisible only by itself and 1. A compositenumber is a natural number greater than 1 that is not prime.

• Thesetofprimenumbers: 52, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, % 6• Thesetofcompositenumbers: 54, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, % 6

Two important subsets of the set of integers are the even and odd integers. The evenintegers are the integers that are exactly divisible by 2. The oddintegers are the integers that are not exactly divisible by 2.

• Thesetofevenintegers: 5 % , 210, 28, 26, 24, 22, 0, 2, 4, 6, 8, 10, % 6 • Thesetofoddintegers:5 % , 29, 27, 25, 23, 21, 1, 3, 5, 7, 9, % 6

So far, we have listed numbers inside braces to specify sets. This method is called the rostermethod. When we give a rule to determine which numbers are in a set, we are using set-buildernotation. To use set-builder notation to denote the set of prime numbers, we write

5x  0  x is a prime number6 Read as “the set of all numbers x such that x is a prime number.” Recall that when a letter stands for a number, it is called a variable.

variable such rule that determines that membership in the set

The fractions of arithmetic are called rational numbers.

Rationalnumbers are fractions that have an integer numerator and a nonzero integer denominator. Using set-builder notation, the rational numbers are

e ab

0 a is an integer and b is a nonzero integer f

Rational numbers can be written as fractions or decimals. Some examples of rational numbers are

5 551

, 34

5 0.75, 213

5 20.333 % , 2511

5 20.454545 % The = sign indicates that two quantities are equal.

▲ ▲ ▲

Caution

Rememberthatthedenominatorofafractioncanneverbe0.

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A4 AppendixAAReviewofBasicAlgebra

real numbers

These examples suggest that the decimal forms of all rational numbers are either terminating decimals or repeating decimals.

eXaMPLe 2 determining whether the decimal Form of a Fraction terminates or repeats

Determine whether the decimal form of each fraction terminates or repeats:

a. 716

b. 6599

sOLUtiOn In each case, we perform a long division and write the quotient as a decimal.

a. To change 716 to a decimal, we perform a long division to get 7

16 5 0.4375. Since 0.4375 terminates, we can write 7

16 as a terminating decimal.

b. To change 6599 to a decimal, we perform a long division to get 65

99 5 0.656565 % . Since 0.656565 ... repeats, we can write 65

99 as a repeating decimal.

We can write repeating decimals in compact form by using an overbar. For example, 0.656565 % 5 0.65.

self Check 2 Determine whether the decimal form of each fraction terminates or repeats:

a. 3899

b. 78

now try exercise 35.

Some numbers have decimal forms that neither terminate nor repeat. These nonterminating, nonrepeating decimals are called irrationalnumbers. Three exam-ples of irrational numbers are

1.010010001000010 % "2 5 1.414213562 % and p 5 3.141592654 %

The union of the set of rational numbers (the terminating and repeating deci-mals) and the set of irrational numbers (the nonterminating, nonrepeating deci-mals) is the set of real numbers (the set of all decimals).

A realnumber is any number that is rational or irrational. Using set-builder notation, the set of real numbers is

5x 0 x is a rational or an irrational number6

eXaMPLe 3 Classifying real numbers

In the set 523, 22, 0, 12, 1, "5, 2, 4, 5, 66, list all

a. even integers b. prime numbers c. rational numbers

sOLUtiOn We will check to see whether each number is a member of the set of even integers, the set of prime numbers, and the set of rational numbers.

a. even integers: –2, 0, 2, 4, 6

b. prime numbers: 2, 5

c. rational numbers: –3, –2, 0, 12, 1, 2, 4, 5, 6

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SectionA.1SetsofRealNumbersA5

Properties of real numbers

self Check 3 In the set in Example 3, list all a. odd integers b. composite numbersc. irrational numbers.

now try exercise 43.

Figure A-2 shows how the previous sets of numbers are related.

, π, 9.911––16

8 – – 5

Irrational numbers

−√5, π, √21, 2√101

2 –,5

13 – ––, 7

Rational numbers

−6, –1.25, 0, , 5 , 80 2 – 3

3 – 4

111 –––, 53

Integers

−4, −1, 0, 21

Negative integers

−47, −17, −5, −1

Natural numbers(positive integers)

1, 12, 38, 990

Whole numbers

0, 1, 4, 8, 10, 53, 101

Zero

0

Noninteger rational numbers

– 0.1, 3.17

−3, , −√2, 0,

Real numbers

FiGUre A-2

2. identify Properties of real numbersWhen we work with real numbers, we will use the following properties.

If a, b, and c are real numbers,

TheCommutativePropertiesforAdditionandMultiplication

a 1 b 5 b 1 a ab 5 ba

TheAssociativePropertiesforAdditionandMultiplication

1a 1 b2 1 c 5 a 1 1b 1 c2 1ab2c 5 a 1bc2TheDistributivePropertyofMultiplicationoverAdditionorSubtraction

a 1b 1 c2 5 ab 1 ac or a 1b 2 c2 5 ab 2 ac

TheDoubleNegativeRule

2 12a2 5 a

WhentheAssociativePropertyisused,theorderoftherealnumbersdoesnotchange.Therealnumbersthatoccurwithintheparentheseschange.

The Distributive Property also applies when more than two terms are within parentheses.

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A6 Appendix A A Review of Basic Algebra

EXAMPLE 4 Identifying Properties of Real Numbers

a. 19 1 22 1 3 5 9 1 12 1 32 b. 3 1x 1 y 1 22 5 3x 1 3y 1 3 ? 2

SOLUTION We will compare the form of each statement to the forms listed in the properties of real numbers box.

a. This form matches the Associative Property of Addition.

b. This form matches the Distributive Property.

Self Check 4

a. mn 5 nm b. 1xy2z 5 x 1yz2 c. p 1 q 5 q 1 p

Now Try Exercise 17.

3. Graph Subsets of Real Numbers on the Number LineWe can graph subsets of real numbers on the number line. The number line shown in Figure A-3 continues forever in both directions. The positive numbers are repre-sented by the points to the right of 0, and the negative numbers are represented by the points to the left of 0.

–5 –4 –3 –2 –1 0 1 2 3 4 5

Negative numbers Positive numbers

FIGURE A-3

Figure A-4(a) shows the graph of the natural numbers from 1 to 5. The point associated with each number is called the graph of the number, and the number is called the coordinate of its point.

Figure A-4(b) shows the graph of the prime numbers that are less than 10.Figure A-4(c) shows the graph of the integers from 24 to 3.Figure A-4(d) shows the graph of the real numbers 27

3, 234, 0.3, and "2.

(a)

–1 0 1 2 3 4 5 6

(b)

0 1 2 3 4 5 6 7 8 9 10

(c)

–5 –4 –3 –2 –1 0 1 2 3 4

–3/4–7/3–

0.3

(d)

–3 –2 –1 0 1 2

2

FIGURE A-4

The graphs in Figure A-4 suggest that there is a one-to-one correspondence be-

tween the set of real numbers and the points on a number line. This means that to each real number there corresponds exactly one point on the number line, and to each point on the number line there corresponds exactly one real-number coordinate.

EXAMPLE 5 Graphing a Set of Numbers on a Number Line

Graph the set U23, 243, 0, "5V.

Comment

Zero is neither positive nor negative.

Comment

"2 can be shown as the diagonal of a square with sides of length 1.

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SectionA.1SetsofRealNumbersA7

sOLUtiOn We will mark (plot) each number on the number line. To the nearest tenth, "5 5 2.2.

–4/3 5

–3 –2 –1 0 1 32

self Check 5 Graph the set U22, 34, "3V. (Hint: To the nearest tenth, "3 5 1.7.)

now try exercise 51.

4. Graph intervals on the number LineTo show that two quantities are not equal, we can use an inequalitysymbol.

Symbol Readas Examples

2 “isnotequalto” 5 2 8 and 0.25 2 13

, “islessthan” 12 , 20 and 0.17 , 1.1

. “isgreaterthan” 15 . 9 and 12 . 0.2

# “islessthanorequalto” 25 # 25 and 1.7 # 2.3

$ “isgreaterthanorequalto” 19 $ 19 and 15.2 $ 13.7

< “isapproximatelyequalto” "2 < 1.414 and "3 < 1.732

It is possible to write an inequality with the inequality symbol pointing in the opposite direction. For example,

• 12 , 20 is equivalent to 20 . 12• 2.3 $ 21.7 is equivalent to 21.7 # 2.3

In Figure A-3, the coordinates of points get larger as we move from left to right on a number line. Thus, if a and b are the coordinates of two points, the one to the right is the greater. This suggests the following facts:

• If a . b, point a lies to the right of point b on a number line.• If a , b, point a lies to the left of point b on a number line.

Figure A-5(a) shows the graph of the inequality x . 22 (or 22 , x). This graph includes all real numbers x that are greater than 22. The parenthesis at 22 indicates that 22 is not included in the graph. Figure A-5(b) shows the graph of x # 5 (or 5 $ x). The bracket at 5 indicates that 5 is included in the graph.

(−2

(a)

]5

(b)

FiGUre A-5

Sometimes two inequalities can be written as a single expression called a compoundinequality. For example, the compound inequality

5 , x , 12

is a combination of the inequalities 5 , x and x , 12. It is read as “5 is less than x, and x is less than 12,” and it means that x is between 5 and 12. Its graph is shown in Figure A-6.

)(5 12

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A8 AppendixAAReviewofBasicAlgebra

The graphs shown in Figures A-5 and A-6 are portions of a number line called intervals. The interval shown in Figure A-7(a) is denoted by the inequality 22 , x , 4, or in intervalnotation as 122, 42 . The parentheses indicate that the endpoints are not included. The interval shown in Figure A-7(b) is denoted by the inequality x . 1, or as 11, `2 in interval notation.

)(−2 4

(−2, 4)−2 < x < 4

x is between −2 and 4.

(a)

1

(1, ∞)x > 1

x is greater than 1.

(b)

(

FiGUre A-7

Thesymbol`(infinity)isnotarealnumber.ItisusedtoindicatethatthegraphinFigureA-7(b)extendsinfinitelyfartotheright.

A compound inequality such as 22 , x , 4 can be written as two separate inequalities:

x . 22 and x , 4

This expression represents the intersection of two intervals. In interval nota-tion, this expression can be written as

122, `2 d 12`, 42 Read the symbol d as “intersection.”

Since the graph of 22 , x , 4 will include all points whose coordinates satisfy both x . 22 and x , 4 at the same time, its graph will include all points that are larger than 22 but less than 4. This is the interval 122, 42 , whose graph is shown in Figure A-7(a).

eXaMPLe 6 Writing an inequality in interval notation and Graphing the inequality

Write the inequality 23 , x , 5 in interval notation and graph it.

sOLUtiOn This is the interval 123, 52 . Its graph includes all real numbers between 23 and 5, as shown in Figure A-8.

)(−3 5

FiGUre A-8

self Check 6 Write the inequality 22 , x # 5 in interval notation and graph it.

now try exercise 63.

If an interval extends forever in one direction, it is called an unboundedinterval.

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SectionA.1SetsofRealNumbersA9

Unbounded intervals

Interval Inequality Graph

1a, `2 x . a (a

12, `2 x . 2 (–1–2 0 1 2 3 4 5 6

3a, `2 x $ a [a

32, `2 x $ 2 [–1–2 0 1 2 3 4 5 6

12`, a2 x , a (a

12`, 22 x , 2 )–3–4 –2 –1 0 1 2 3 4

12`, a 4 x # a [a

12`, 2 4 x # 2 ]–3–4 –2 –1 0 1 2 3 4

12`, `2 2` , x , `

A bounded interval with no endpoints is called an openinterval. Figure A-9(a) shows the open interval between 23 and 2. A bounded interval with one endpoint is called a half-openinterval. Figure A-9(b) shows the half-open interval between 22 and 3, including 22.

Intervals that include two endpoints are called closed intervals. Figure A-10 shows the graph of a closed interval from 22 to 4.

Open intervals, half-Open intervals, and Closed intervals

Interval Inequality Graph

Open

1a, b2 a , x , b )(a b

122, 32 22 , x , 3 )(–3–4 –2 –1 0 1 2 3 4 5

Half-Open

3a, b2 a # x , b )[a b

322, 32 22 # x , 3 )[–3–4 –2 –1 0 1 2 3 4 5

Half-Open

1a, b 4 a , x # b ](a b

122, 3 4 22 , x # 3 ](–3–4 –2 –1 0 1 2 3 4 5

Closed

3a, b 4 a # x # b ][a b

322, 3 4 22 # x # 3 ][–3–4 –2 –1 0 1 2 3 4 5

)[−2 3

[−2, 3)−2 ≤ x < 3

(b)

)(−3 2

(−3, 2)−3 < x < 2

(a)

FiGUre A-9

][−2 4

[−2, 4]−2 ≤ x ≤ 4

FiGUre A-10

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A10 AppendixAAReviewofBasicAlgebra

eXaMPLe 7 Writing an inequality in interval notation and Graphing the inequality

Write the inequality 3 # x in interval notation and graph it.

sOLUtiOn The inequality 3 # x can be written in the form x $ 3. This is the interval 33, `2 . Its graph includes all real numbers greater than or equal to 3, as shown in Figure A-11.

[3

FiGUre A-11

self Check 7 Write the inequality 5 . x in interval notation and graph it.

now try exercise 65.

eXaMPLe 8 Writing an inequality in interval notation and Graphing the inequality

Write the inequality 5 $ x $ 21 in interval notation and graph it.

sOLUtiOn The inequality 5 $ x $ 21 can be written in the form

21 # x # 5

This is the interval 321, 5 4. Its graph includes all real numbers from 21 to 5. The graph is shown in Figure A-12.

][−1 5

FiGUre A-12

self Check 8 Write the inequality 0 # x # 3 in interval notation and graph it.

now try exercise 69.

The expression

x , 22 or x $ 3 Read as “x is less than 22 or x isgreaterthanorequalto3.”

represents the union of two intervals. In interval notation, it is written as

12`, 222 c 33, `2 Read the symbol c as “union.”

Its graph is shown in Figure A-13.

[3

)−2

FiGUre A-13

5. define absolute ValueThe absolutevalue of a real number x (denoted as 0 x 0 ) is the distance on a number line between 0 and the point with a coordinate of x. For example, points with coordinates of 4 and 24 both lie four units from 0, as shown in Figure A-14. Therefore, it follows that

024 0 5 0 4 0 5 4

–1–2–3–4–5 0 1 2 3 4 5

4 units4 units

FiGUre A-14

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SectionA.1SetsofRealNumbersA11

absolute Value

In general, for any real number x,

02x 0 5 0 x 0We can define absolute value algebraically as follows.

If x is a real number, then

0 x 0 5 x when x $ 0

0 x 0 5 2x when x , 0

This definition indicates that when x is positive or 0, then x is its own absolute value. However, when x is negative, then 2x (which is positive) is its absolute value. Thus, 0 x 0 is always nonnegative.

0 x 0 $ 0 for all real numbers x

eXaMPLe 9 Using the definition of absolute Value

Write each number without using absolute value symbols:

a. 0 3 0 b. 024 0 c. 0 0 0 d. 2 028 0 sOLUtiOn In each case, we will use the definition of absolute value.

a. 0 3 0 5 3 b. 024 0 5 4 c. 0 0 0 5 0 d. 2 028 0 5 2 182 5 28

self Check 9 Write each number without using absolute value symbols:a. 0210 0 b. 0 12 0 c. 2 0 6 0 now try exercise 85.

In Example 10, we must determine whether the number inside the absolute value is positive or negative.

eXaMPLe 10 simplifying an expression with absolute Value symbols

Write each number without using absolute value symbols:

a. 0p 2 1 0 b. 0 2 2 p 0 c. 0 2 2 x 0 if x $ 5

sOLUtiOn a. Since p < 3.1416, p 2 1 is positive, and p 2 1 is its own absolute value.

0p 2 1 0 5 p 2 1

b. Since 2 2 p is negative, its absolute value is 2 12 2 p2 .0 2 2 p 0 5 2 12 2 p2 5 22 2 1 2 p2 5 22 1 p 5 p 2 2

c. Since x $ 5, the expression 2 2 x is negative, and its absolute value is 2 12 2 x2 .0 2 2 x 0 5 2 12 2 x2 5 22 1 x 5 x 2 2 provided x $ 5

self Check 10 Write each number without using absolute value symbols. 1Hint: "5 < 2.236.2a. P 2 2 "5 P b. 0 2 2 x 0 if x # 1

now try exercise 89.

Caution

Rememberthatxisnotalwayspositiveand2xisnotalwaysnegative.

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A12 AppendixAAReviewofBasicAlgebra

distance between two Points

6. Find distances on the number LineOn the number line shown in Figure A-15, the distance between the points with coordinates of 1 and 4 is 4 2 1, or 3 units. However, if the subtraction were done in the other order, the result would be 1 2 4, or 23 units. To guarantee that the distance between two points is always positive, we can use absolute value symbols. Thus, the distance d between two points with coordinates of 1 and 4 is

d 5 0 4 2 1 0 5 0 1 2 4 0 5 3

–1 0 1 2 3 4 5

d = |4 − 1| = 3

FiGUre A-15

In general, we have the following definition for the distance between two points on the number line.

If a and b are the coordinates of two points on the number line, the distance between the points is given by the formula

d 5 0 b 2 a 0

eXaMPLe 11 Finding the distance between two Points on a number Line

Find the distance on a number line between points with coordinates of a. 3 and 5 b. 22 and 3 c. 25 and 21

sOLUtiOn We will use the formula for finding the distance between two points.

a. d 5 0 5 2 3 0 5 0 2 0 5 2

b. d 5 0 3 2 1222 0 5 0 3 1 2 0 5 0 5 0 5 5

c. d 5 021 2 1252 0 5 021 1 5 0 5 0 4 0 5 4

self Check 11 Find the distance on a number line between points with coordinates of a. 4 and 10 b. 22 and 27

now try exercise 99.

self Check answers 1. a. no b. {a, c, d, e, i, o, u} c. {a, e, i} 2. a. repeats

b. terminates 3. a. 23, 1, 5 b. 4, 6 c. "5 4. a. Commutative

Property of Multiplication b. Associative Property of Multiplication

c. Commutative Property of Addition 5. 3/4–2 3

–3 –2 –1 0 1 2

6. 122, 5 4 ](−2 5

7. 12`, 52 )5

8. 30, 3 4 ][0 3

9. a. 10 b. 12 c. 26

10. a. "5 2 2 b. 2 2 x 11. a. 6 b. 5

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SectionA.1SetsofRealNumbersA13

Exercises A.1Getting readyYou should be able to complete these vocabulary and concept statements before you proceed to the practice exercises.

Fill in the blanks. 1. A is a collection of objects.

2. If every member of one set B is also a member of a second set A, then B is called a of A.

3. If A and B are two sets, the set that contains all members that are in sets A and B or both is called the of A and B.

4. If A and B are two sets, the set that contains all members that are in both sets is called the of A and B.

5. A real number is any number that can be expressed as a .

6. A is a letter that is used to represent a number.

7. The smallest prime number is .

8. All integers that are exactly divisible by 2 are called integers.

9. Natural numbers greater than 1 that are not prime are called numbers.

10. Fractions such as 23, 8

2, and 279 are called

numbers.

11. Irrational numbers are that don’t termi-nate and don’t repeat.

12. The symbol is read as “is less than or equal to.”

13. On a number line, the numbers are to the left of 0.

14. The only integer that is neither positive nor negative is .

15. The Associative Property of Addition states that 1x 1 y2 1 z 5 .

16. The Commutative Property of Multiplication states that xy 5 .

17. Use the Distributive Property to complete the state-ment: 5 1m 1 22 5 .

18. The statement 1m 1 n2p 5 p 1m 1 n2 illustrates the Property of .

19. The graph of an is a portion of a number line.

20. The graph of an open interval has endpoints.

21. The graph of a closed interval has endpoints.

22. The graph of a interval has one endpoint.

23. Except for 0, the absolute value of every number is .

24. The between two distinct points on a number line is always positive.

LetN 5 the set of natural numbers

W 5 the set of whole numbers

Z 5 the set of integers

Q 5 the set of rational numbers

R 5 the set of real numbers

Determine whether each statement is true or false. Read the symbol ( as “is a subset of.”25. N ( W 26. Q ( R

27. Q ( N 28. Z ( Q

29. W ( Z 30. R ( Z

PracticeLet A 5 5a,b,c,d,e6, B 5 1d,e,f,g6, and C 5 5a,c,e,f6. Find each set.

31. A c B 32. A d B

33. A d C 34. B c C

Determine whether the decimal form of each fraction terminates or repeats.

35. 916

36.38

37. 311

38.512

Consider the following set: 525,24,22

3,0,1,"2,2,2.75,6,76 .

39. Which numbers are natural numbers?

40. Which numbers are whole numbers?

41. Which numbers are integers?

42. Which numbers are rational numbers?

43. Which numbers are irrational numbers?

44. Which numbers are prime numbers?

45. Which numbers are composite numbers?

46. Which numbers are even integers?

47. Which numbers are odd integers?

48. Which numbers are negative numbers?

Graph each subset of the real numbers on a number line.49. The natural numbers between 1 and 5

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A14 AppendixAAReviewofBasicAlgebra

50. The composite numbers less than 10

51. The prime numbers between 10 and 20

52. The integers from –2 to 4

53. The integers between –5 and 0

54. The even integers between –9 and –1

55. The odd integers between –6 and 4

56. 20.7, 1.75, and 378

Write each inequality in interval notation and graph the interval.

57. x . 2 58. x , 4

59. 0 , x , 5 60. 22 , x , 3

61. x . 24 62. x , 3

63. 22 # x , 2 64. 24 , x # 1

65. x # 5 66. x $ 21

67. 25 , x # 0 68. 23 # x , 4

69. 22 # x # 3 70. 24 # x # 4

71. 6 $ x $ 2 72. 3 $ x $ 22

Write each pair of inequalities as the intersection of two intervals and graph the result.

73. x . 25 and x , 4

74. x $ 23 and x , 6

75. x $ 28 and x # 23

76. x . 1 and x # 7

Write each inequality as the union of two intervals and graph the result. 77. x , 22 or x . 2

78. x # 25 or x . 0

79. x # 21 or x $ 3

80. x , 23 or x $ 2

Write each expression without using absolute value symbols.

81. 0 13 0 82. 0217 0 83. 0 0 0 84. 2 0 63 0

85. 2 028 0 86. 0225 0 87. 2 0 32 0 88. 2 026 0 89. 0p 2 5 0 90. 0 8 2 p 0

91. 0p 2 p 0 92. 0 2p 0 93. 0 x 1 1 0 and x $ 2 94. 0 x 1 1 0 and x # 2 2

95. 0 x 2 4 0 and x , 0 96. 0 x 2 7 0 and x . 10

Find the distance between each pair of points on the number line. 97. 3 and 8 98. 25 and 12

99. 28 and 23 100. 6 and 220

applications101. What subset of the real numbers would you use to

describe the populations of several cities?

102. What subset of the real numbers would you use to describe the subdivisions of an inch on a ruler?

103. What subset of the real numbers would you use to report temperatures in several cities?

104. What subset of the real numbers would you use to describe the financial condition of a business?

discovery and Writing105. Explain why 2x could be positive.

106. Explain why every integer is a rational number.

107. Is the statement 0 ab 0 5 0 a 0 ? 0 b 0 always true? Explain.

108. Is the statement ` ab` 5

0 a 00 b 0 1b 2 02 always true?

Explain.

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SectionA.2IntegerExponentsandScientificNotationA15

natural-number exponents

109. Is the statement 0 a 1 b 0 5 0 a 0 1 0 b 0 always true? Explain.

111. Explain why it is incorrect to write a , b . c if a , b and b . c.

110. Under what conditions will the statement given in Exercise 109 be true?

112. Explain why 0 b 2 a 0 5 0 a 2 b 0 .

A.2 Integer Exponents and Scientific NotationInthissection,wewilllearnto

1.Definenatural-numberexponents.2.Applytherulesofexponents.3.Applytherulesfororderofoperationstoevaluateexpressions.4.Expressnumbersinscientificnotation.5.Usescientificnotationtosimplifycomputations.

The number of cells in the human body is approximated to be one hundred tril-lion or 100,000,000,000,000. One hundred trillion is 1102 1102 1102 # # # 1102 , where ten occurs fourteen times. Fourteen factors of ten can be written as 1014.

In this section, we will use integer exponents to represent repeated multiplica-tion of numbers.

1. define natural-number exponentsWhen two or more quantities are multiplied together, each quantity is called a factor of the product. The exponential expression x4 indicates that x is to be used as a factor four times.

4 factors of x

x4 5 x ? x ? x ? x

In general, the following is true.

For any natural number n,

n factors of x

xn 5 x ? x ? x ? c ? x

In the exponentialexpression xn, x is called the base, and n is called the exponent or the power to which the base is raised. The expression xn is called a powerof x. From the definition, we see that a natural-number exponent indicates how many times the base of an exponential expression is to be used as a factor in a product. If an exponent is 1, the 1 is usually not written:

x1 5 x

eXaMPLe 1 Using the definition of natural-number exponents

Write each expression without using exponents:

a. 42 b. 1242 2 c. 253 d. 1252 3 e. 3x4 f. 13x2 4

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A16 AppendixAAReviewofBasicAlgebra

aCCent On teChnOLOGy

sOLUtiOn In each case, we apply the definition of natural-number exponents.

a. 42 5 4 ? 4 5 16 Read 42 as “four squared.”

b. 1242 2 5 1242 1242 5 16 Read 1242 2 as “negative four squared.”

c. 253 5 25 152 152 5 2125 Read 253 as “the negative of five cubed.”

d. 1252 3 5 1252 1252 1252 5 2125 Read 1252 3 as “negative five cubed.”

e. 3x4 5 3 ? x ? x ? x ? x Read 3x4 as “3 times x to the fourth power.”

f. 13x2 4 5 13x2 13x2 13x2 13x2 5 81 ? x ? x ? x ? x Read 13x2 4 as “3x to the fourth power.”

self Check 1 Write each expression without using exponents:a. 73 b. 1232 2 c. 5a3 d. 15a2 4

now try exercise 19.

Notethedistinctionbetweenaxnand 1ax2 n:

n factors of x n factors of ax

axn 5 a ? x ? x ? x ? c ? x 1ax2n 5 1ax2 1ax2 1ax2 ? c ? 1ax2

Alsonotethedistinctionbetween2xnand 12x2 n:

n factors of x n factors of 2x

2xn 5 2 1x ? x ? x ? c ? x2 12x2 n 5 12x2 12x2 12x2 ? c ? 12x2

UsingaCalculatortoFindPowers We can use a graphing calculator to find powers of numbers. For example, consider 2.353.

• Input 2.35 and press the key.• Input 3 and press ENTER .

FiGUre A-16

We see that 2.353 5 12.977875 as the figure shows above.

2. apply the rules of exponentsWe begin the review of the rules of exponents by considering the product xmxn. Since xm indicates that x is to be used as a factor m times, and since xn indicates that x is to be used as a factor n times, there are m 1 n factors of x in the product xmxn.

m 1 n factors of x

m factors of x n factors of x

xmxn 5 x ? x ? x ? c ? x ? x ? x ? x ? c ? x 5 xm1n

Comment

Comment

Tofindpowersonascientificcalculatorusetheyxkey.

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SectionA.2IntegerExponentsandScientificNotationA17

Power rules of exponents

Product rule for exponents

This suggests that to multiply exponential expressions with the same base, we keep the base and add the exponents.

If m and n are natural numbers, then

xmxn 5 xm1n

To find another property of exponents, we consider the exponential expression 1xm2 n. In this expression, the exponent n indicates that xm is to be used as a factor n times. This implies that x is to be used as a factor mn times.

mn factors of x

n factors of xm

1xm2 n 5 1xm2 1xm2 1xm2 ? c ? 1xm2 5 xmn

This suggests that to raise an exponential expression to a power, we keep the base and multiply the exponents.

To raise a product to a power, we raise each factor to that power.

n factors of xy n factors of x n factors of y

1xy2 n 5 1xy2 1xy2 1xy2 ? c ? 1xy2 5 1x ? x ? x ? c ? x2 1y ? y ? y ? c ? y2 5 xnyn

To raise a fraction to a power, we raise both the numerator and the denomina-tor to that power. If y 2 0, then

n factors of xy

axyb

n

5 axyb ax

yb ax

yb ? c ? ax

yb

n factors of x

5xxx ? c ? xyyy ? c ? y

n factors of y

5xn

yn

The previous three results are called the Power Rules of Exponents.

If m and n are natural numbers, then

1xm2 n 5 xmn 1xy2 n 5 xnyn axyb

n

5xn

yn 1y 2 02

eXaMPLe 2 Using exponent rules to simplify expressions with natural-number exponents

Simplify: a. x5x7 b. x2y3x5y c. 1x42 9 d. 1x2x52 3

e. a xy2b

5

f. a5x2yz3 b

2

Comment

TheProductRuleappliestoexponentialexpressionswiththesamebase.Aproductoftwopowerswithdifferentbases,suchasx4y3,cannotbesimplified.

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A18 AppendixAAReviewofBasicAlgebra

negative exponents

Zero exponent

sOLUtiOn In each case, we will apply the appropriate rule of exponents.

a. x5x7 5 x517 5 x12 b. x2y3x5y 5 x215y311 5 x7y4

c. 1x42 9 5 x4?9 5 x36 d. 1x2x52 3 5 1x7 2 3 5 x21

e. a xy2b

5

5x5

1y22 5 5x5

y10 1y 2 02

f. a5x2yz3 b

2

552 1x22 2y2

1z32 2 525x4y2

z6 1z 2 02

self Check 2 Simplify:

a. 1y32 2 b. 1a2a42 3 c. 1x22 3 1x32 2 d. a3a3b2

c3 b3

1c 2 02

now try exercise 49.

If we assume that the rules for natural-number exponents hold for exponents of 0, we can write

x0xn 5 x01n 5 xn 5 1xn

Since x0xn 5 1xn, it follows that if x 2 0, then x0 5 1.

x0 5 1 1x 2 02

If we assume that the rules for natural-number exponents hold for exponents that are negative integers, we can write

x2nxn 5 x2n1n 5 x0 5 1 1x 2 02However, we know that

1xn ? xn 5 1 1x 2 02 1

xn ? xn 5xn

xn , and any nonzero number divided by itself is 1.

Since x2nxn 5 1xn ? xn, it follows that x2n 5 1

xn 1x 2 02 .

If n is an integer and x 2 0, then

x2n 51xn and

1x2n 5 xn

Because of the previous definitions, all of the rules for natural-number expo-nents will hold for integer exponents.

eXaMPLe 3 simplifying expressions with integer exponents

Simplify and write all answers without using negative exponents:

a. 13x2 0 b. 3 1x02 c. x24 d. 1

x26 e. x23x f. 1x24x8225

sOLUtiOn We will use the definitions of zero exponent and negative exponents to simplify each expression.

a. 13x2 0 5 1 b. 3 1x02 5 3 112 5 3 c. x24 51x4 d.

1x26 5 x6

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SectionA.2IntegerExponentsandScientificNotationA19

Quotient rule for exponents

e. x23x 5 x2311 f. 1x24x8225 5 1x4225

5 x22 5 x220

51x2 5

1x20

self Check 3 Simplify and write all answers without using negative exponents:

a. 7a0 b. 3a22 c. a24a2 d. 1a3a272 3

now try exercise 59.

To develop the Quotient Rule for Exponents, we proceed as follows:

xm

xn 5 xma 1xnb 5 xmx2n 5 xm1 12n2 5 xm2n 1x 2 02

This suggests that to divide two exponential expressions with the same nonzero base, we keep the base and subtract the exponent in the denominator from the expo-nent in the numerator.

If m and n are integers, then

xm

xn 5 xm2n 1x 2 02

eXaMPLe 4 simplifying expressions with integer exponents

Simplify and write all answers without using negative exponents:

a. x8

x5 b. x2x4

x25

sOLUtiOn We will apply the Product and Quotient Rules of Exponents.

a. x8

x5 5 x825 b. x2x4

x25 5x6

x25

5 x3 5 x62 1252

5 x11

self Check 4 Simplify and write all answers without using negative exponents:

a. x26

x2 b. x4x23

x2

now try exercise 69.

eXaMPLe 5 simplifying expressions with integer exponents

Simplify and write all answers without using negative exponents:

a. ax3y22

x22y3b22

b. axyb

2n

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A20 AppendixAAReviewofBasicAlgebra

strategy for evaluating expressions Using Order of Operations

a Fraction to a negative Power

sOLUtiOn We will apply the appropriate rules of exponents.

a. ax3y22

x22y3b22

5 1x32 1222y2223222

5 1x5y25222

5 x210y10

5y10

x10

b. axyb

2n

5x2n

y2n

5x2nxnyn

y2nxnyn Multiply numerator and denominator by 1 in the form xnyn

xnyn.

5x0yn

y0xn x2nxn 5 x0 and y2nyn 5 y0.

5yn

xn x0 5 1 and y0 5 1.

5 ayxb

n

self Check 5 Simplify and write all answers without using negative exponents:

a. ax4y23

x23y2b2

b. a2a3b

b23

now try exercise 75.

Part b of Example 5 establishes the following rule.

If n is a natural number, then

axyb

2n

5 ayxb

n

(x 2 0 and y 2 0)

3. apply the rules for Order of Operations to evaluate expressions

When several operations occur in an expression, we must perform the operations in the following order to get the correct result.

If an expression does not contain grouping symbols such as parentheses or brackets, follow these steps:

1. Find the values of any exponential expressions. 2. Perform all multiplications and/or divisions, working from left to right. 3. Perform all additions and/or subtractions, working from left to right.

• If an expression contains grouping symbols such as parentheses, brackets, or braces, use the rules above to perform the calculations within each pair of grouping symbols, working from the innermost pair to the outermost pair.

• In a fraction, simplify the numerator and the denominator of the fraction separately. Then simplify the fraction, if possible.

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SectionA.2IntegerExponentsandScientificNotationA21

ManystudentsremembertheOrderofOperationsRulewiththeacronymPEMDAS:

• Parentheses• Exponents• Multiplication• Division• Addition• Subtraction

For example, to simplify 3 34 2 16 1 102 4

22 2 16 1 72 , we proceed as follows:

3 34 2 16 1 102 4

22 2 16 1 72 53 14 2 162

22 2 16 1 72 Simplify within the inner parentheses: 6 1 10 5 16.

53 1212222 2 13

Simplify within the parentheses: 4 2 16 5 212, and 6 1 7 5 13.

53 121224 2 13

Evaluate the power: 22 5 4.

523629

3 12122 5 236. 4 2 13 5 29.

5 4

eXaMPLe 6 evaluating algebraic expressions

If x 5 22, y 5 3, and z 5 24, evaluate

a. 2x2 1 y2z b. 2z3 2 3y2

5x2

sOLUtiOn In each part, we will substitute the numbers for the variables, apply the rules of order of operations, and simplify.

a. 2x2 1 y2z 5 2 1222 2 1 32 1242 5 2 142 1 9 1242 Evaluate the powers.

5 24 1 12362 Do the multiplication.

5 240 Do the addition.

b. 2z3 2 3y2

5x2 52 1242 3 2 3 132 2

5 1222 2

52 12642 2 3 192

5 142 Evaluate the powers.

52128 2 27

20 Do the multiplications.

52155

20 Do the subtraction.

5 2314

Simplify the fraction.

self Check 6 If x 5 3 and y 5 22, evaluate 2x2 2 3y2

x 2 y.

now try exercise 91.

4. express numbers in scientific notationScientists often work with numbers that are very large or very small. These numbers can be written compactly by expressing them in scientific notation.

Comment

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A22 AppendixAAReviewofBasicAlgebra

scientific notation A number is written in scientific notation when it is written in the form

N 3 10n

where 1 # 0N 0 , 10 and n is an integer.

Light travels 29,980,000,000 centimeters per second.To express this number in scientific notation, we must write it as the product of

a number between 1 and 10 and some integer power of 10. The number 2.998 lies between 1 and 10. To get 29,980,000,000, the decimal point in 2.998 must be moved ten places to the right. This is accomplished by multiplying 2.998 by 1010.

Standard notation 29,980,000,000 5 2.998 3 1010 Scientific notation

One meter is approximately 0.0006214 mile. To express this number in scientific notation, we must write it as the product of a number between 1 and 10 and some integer power of 10. The number 6.214 lies between 1 and 10. To get 0.0006214, the decimal point in 6.214 must be moved four places to the left. This is accomplished by multiplying 6.214 by 1

104 or by multiplying 6.214 by 1024.

Standard notation 0.0006214 5 6214 3 1024 Scientific notation

To write each of the following numbers in scientific notation, we start to the right of the first nonzero digit and count to the decimal point. The exponent gives the number of places the decimal point moves, and the sign of the exponent indi-cates the direction in which it moves.

a. 3 7 2 0 0 0 5 3.72 3 105 5 places to the right.

b. 0 . 0 0 0 5 3 7 5 5.37 3 1024 4 places to the left.

c. 7.36 5 7.36 3 100 No movement of the decimal point.

eXaMPLe 7 Writing numbers in scientific notation

Write each number in scientific notation: a. 62,000 b. 20.0027

sOLUtiOn a. We must express 62,000 as a product of a number between 1 and 10 and some integer power of 10. This is accomplished by multiplying 6.2 by 104.

62,000 5 6.2 3 104

b. We must express 20.0027 as a product of a number whose absolute value is between 1 and 10 and some integer power of 10. This is accomplished by multi-plying 22.7 by 1023.

20.0027 5 22.7 3 1023

self Check 7 Write each number in scientific notation:a. 293,000,000 b. 0.0000087

now try exercise 103.

eXaMPLe 8 Writing numbers in standard notation

Write each number in standard notation:

a. 7.35 3 102 b. 3.27 3 1025

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▲

▲

▲

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SectionA.2IntegerExponentsandScientificNotationA23

aCCent On teChnOLOGy

sOLUtiOn a. The factor of 102 indicates that 7.35 must be multiplied by 2 factors of 10. Because each multiplication by 10 moves the decimal point one place to the right, we have

7.35 3 102 5 735

b. The factor of 1025 indicates that 3.27 must be divided by 5 factors of 10. Because each division by 10 moves the decimal point one place to the left, we have

3.27 3 1025 5 0.0000327

self Check 8 Write each number in standard notation: a. 6.3 3 103 b. 9.1 3 1024

now try exercise 111.

5. Use scientific notation to simplify ComputationsAnother advantage of scientific notation becomes evident when we multiply and divide very large and very small numbers.

eXaMPLe 9 Using scientific notation to simplify Computations

Use scientific notation to calculate 13,400,0002 10.000022

170,000,000.

sOLUtiOn After changing each number to scientific notation, we can do the arithmetic on the numbers and the exponential expressions separately.

13,400,0002 10.000022

170,000,0005

13.4 3 1062 12.0 3 102521.7 3 108

56.81.7

3 1061 125228

5 4.0 3 1027

5 0.0000004

self Check 9 Use scientific notation to simplify 1192,0002 10.0015210.00322 14,5002 .

now try exercise 119.

ScientificNotationGraphing calculators will often give an answer in scientific notation. For example, consider 218.

FiGUre A-17

We see in the figure that the answer given is 3.782285936E10, which means3.782285936 3 1010.Not For Sale

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A24 Appendix A A Review of Basic Algebra

To calculate an expression like 218

0.000000000061 on a scientific calculator, it is necessary to convert

the denominator to scientific notation because the number has too many digits to fit on the screen.

• For scientific notation, we enter these numbers and press these keys: 6.1 EXP 11 1/2 .

• To evaluate the expression above, we enter these numbers and press these keys:

21 yx 8 5 4 6.1 EXP 11 1/2 5

The display will read 6.20046874820 . In standard notation, the answer is approximately 620,046,874,800,000,000,000.

Self Check Answers 1.  a. 7 ? 7 ? 7 5 343 b. 1232 1232 5 9 c. 5 ? a ? a ? a d. 15a2 15a2 15a2 15a2 5 625 ? a ? a ? a ? a 2.  a. y6 b. a18 c. x12

    d. 27a9b6

c9 3.  a. 7 b. 3a2 c.

1a2 d.

1a12 4.  a.

1x8 b.

1x

    5.  a. x14

y10 b. 27b3

8a3 6. 65

7.  a. 29.3 3 107 b. 8.7 3 1026

8.  a. 6,300 b. 0.00091 9. 20

Comment

Exercises A.2Getting ReadyYou should be able to complete these vocabulary and concept statements before you proceed to the practice exercises.

Fill in the blanks. 1. Each quantity in a product is called a of

the product.

2. A number exponent tells how many times a base is used as a factor.

3. In the expression 12x2 3, is the exponent and is the base.

4. The expression xn is called an expres-sion.

5. A number is in notation when it is writ-ten in the form N 3 10n, where 1 # 0N 0 , 10 and n is an .

6.  Unless indicate otherwise, are performed before additions.

Complete each exponent rule. Assume x 2 0.

7. xmxn 5 8. 1xm2 n 5

9. 1xy2 n 5 10. xm

xn 5

11. x0 5 12. x2n 5

PracticeWrite each number or expression without using exponents.

13. 132 14. 103

15. 252 16. 1252 2

17. 4x3 18. 14x2 3

19. 125x2 4 20. 26x2 21. 28x4 22.  128x2 4

Write each expression using exponents.

23. 7xxx 24. 28yyyy

25. 12x2 12x2 26. 12a2 12a2 12a2 27. 13t2 13t2 123t2 28. 2 12b2 12b2 12b2 12b2 29. xxxyy 30. aaabbbb

Use a calculator to simplify each expression.

31. 2.23 32. 7.14

33. 20.54 34. 120.22 4

Simplify each expression. Write all answers without using negative exponents. Assume that all variables are restricted to those numbers for which the expression is defined.

35. x2x3 36. y3y4

37. 1z22 3 38. 1t62 7

39. 1y5y22 3 40. 1a3a62a4

41. 1z22 3 1z42 5 42. 1t32 4 1t52 2

43. 1a22 3 1a42 2 44. 1a22 4 1a32 3

45. 13x2 3 46. 122y2 4

47. 1x2y2 3 48. 1x3z42 6

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SectionA.2IntegerExponentsandScientificNotationA25

49. aa2

bb

3

50. a xy3b

4

51. 12x2 0 52. 4x0

53. 14x2 0 54. 22x0

55. z24 56. 1

t22

57. y22y23 58. 2m22m3

59. 1x3x24222 60. 1y22y3224

61. x7

x3 62. r5

r2

63. a21

a17 64. t13

t4

65. 1x22 2

x2x 66.

s9s3

1s22 2

67. am3

n2 b3

68. a t4

t3b3

69. 1a3222

aa2 70. r9r23

1r222 3

71. aa23

b21b24

72. a t24

t23b22

73. ar4r26

r3r23b2

74. 1x23x22 2

1x2x25223

75. ax5y22

x23y2b4

76. ax27y5

x7y24b3

77. a5x23y22

3x2y23 b22

78. a 3x2y25

2x22y26b23

79. a3x5y23

6x25y3b22

80. a12x24y3z25

4x4y23z5 b3

81. 1822z23y221

15y2z222 3 15yz22221

82. 1m22n3p4222 1mn22p32 4

1mn22p3224 1mn2p221

Simplify each expression.

83. 25 362 1 19 2 52 4

4 12 2 32 2 84. 6 33 2 14 2 72 2 4

25 12 2 422

Let x 5 22, y 5 0, and z 5 3 and evaluate each expression. 85. x2 86. 2x2

87. x3 88. 2x3

89. 12xz2 3 90. 2xz3

91. 2 1x2z32z2 2 y2 92.

z2 1x2 2 y22x3z

93. 5x2 2 3y3z 94. 3 1x 2 z2 2 1 2 1y 2 z2 3

95. 23x23z22

6x2z23 96. 125x2z232 2

5xz22

Express each number in scientific notation. 97. 372,000 98. 89,500

99. 2177,000,000 100. 223,470,000,000

101. 0.007 102. 0.00052

103. 20.000000693 104. 20.000000089

105. one trillion 106. one millionth

Express each number in standard notation.

107. 9.37 3 105 108. 4.26 3 109

109. 2.21 3 1025 110. 2.774 3 1022

111. 0.00032 3 104 112. 9,300.0 3 1024

113. 23.2 3 1023 114. 27.25 3 103

Use the method of Example 9 to do each calculation. Write all answers in scientific notation.

115. 165,0002 145,0002

250,000

116. 10.0000000452 10.000000122

45,000,000

117. 10.000000352 1170,0002

0.00000085

118. 10.00000001442 112,0002

600,000

119. 145,000,000,0002 1212,0002

0.00018

120. 10.000000002752 14,7502

500,000,000,000

applications Use scientific notation to compute each answer. Write all answers in scientific notation.

121. Speedofsound The speed of sound in air is 3.31 3 104 centimeters per second. Compute the speed of sound in meters per minute.

122. Volumeofabox Calculate the volume of a box that has dimensions of 6,000 by 9,700 by 4,700 millimeters.

123. Massofaproton The mass of one proton is 0.00000000000000000000000167248 gram. Find the mass of one billion protons.

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A26 AppendixAAReviewofBasicAlgebra

124. Speedoflight The speed of light in a vacuum is approximately 30,000,000,000 centimeters per second. Find the speed of light in miles per hour. 1160,934.4 cm 5 1 mile.2

125. Astronomy The distance d, in miles, of the nth planet from the sun is given by the formula

d 5 9,275,200 33 12n222 1 4 4 To the nearest million miles, find the distance of

Earth and the distance of Mars from the sun. Give each answer in scientific notation.

Jupiter

Mars

Earth

Venus

Mercury

126. LicensePlates The number of different license plates of the form three digits followed by three letters, as in the illustration, is 10 ? 10 ? 10 ? 26 ? 26 ? 26. Write this expression using exponents. Then evaluate it and express the result in scientific notation.

123ABCUTAHWB

COUNTY 12

discovery and Writing 127. Newwaytothecenteroftheearth The spectacular

“blue marble’ image is the most detailed true-color image of the entire Earth to date. A new NASA-developed technique estimates Earth’s center of

mass within 1 millimeter (0.04 inch) a year by using a combination of four space-based techniques.

The distance from the Earth’s center to the North Pole (the polarradius) measures approximately 6,356.750 km, and the distance from the center to the equator (the equatorialradius ) measures approximately 6,378.135 km. Express each distance using scientific notation.

128. Refer to Exercise 127. Given that 1 km is approximately equal to 0.62 miles, use scientific notation to express each distance in miles.

Write each expression with a single base.

129. xnx2 130. xm

x3

131. xmx2

x3 132. x3m15

x2

133. xm11x3 134. an23a3

135. Explain why 2x4 and 12x2 4 represent different numbers.

136. Explain why 32 3 102 is not in scientific notation.

review 137. Graph the interval 122,42 .

138. Graph the interval 12`,23 4 c 33,`2 .

139. Evaluate 0p 2 5 0 . 140. Find the distance between 27 and 25 on the

number line.

NAS

A Go

ddar

d Sp

ace

Flig

ht C

ente

r

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SectionA.3RationalExponentsandRadicalsA27

rational exponents

A.3 Rational Exponents and RadicalsInthissection,wewilllearnto

1.Definerationalexponentswhosenumeratorsare1.2.Definerationalexponentswhosenumeratorsarenot1.3.Defineradicalexpressions.4.Simplifyandcombineradicals.5.Rationalizedenominatorsandnumerators.

“Dead Man’s Curve” is a 1964 hit song by the rock and roll duo Jan Berry and Dean Torrence. The song details a teenage drag race that ends in an accident. Today, dead man’s curve is a commonly used expression given to dangerous curves on our roads. Every curve has a “critical speed.” If we exceed this speed, regardless of how skilled a driver we are, we will lose control of the vehicle.

The radical expression 3.9"r gives the critical speed in miles per hour when we travel a curved road with a radius of r feet. A knowledge of square roots and radicals is important and used in the construction of safe highways and roads. We will study the topic of radicals in this section.

1. define rational exponents Whose numerators are 1 If we apply the rule 1xm2 n 5 xmn to 1251/22 2, we obtain

1251/22 2 5 2511/222 Keep the base and multiply the exponents.

5 251 12 ? 2 5 1

5 25

Thus, 251/2 is a real number whose square is 25. Although both 52 5 25 and 1252 2 5 25, we define 251/2 to be the positive real number whose square is 25:

251/2 5 5 Read 251/2 as “the square root of 25.”

In general, we have the following definition.

If a $ 0 and n is a natural number, then a1/n (read as “the nth root of a”) is the nonnegative real number b such that

bn 5 a

Since b 5 a1/n, we have bn 5 1a1/n2 n 5 a.

eXaMPLe 1 simplifying expressions with rational exponents

In each case, we will apply the definition of rational exponents.

a. 161/2 5 4 Because 42 5 16. Read 161/2 as “the square root of 16.”

b. 271/3 5 3 Because 33 5 27. Read 271/3 as “the cube root of 27.”

c. a 181

b1/4

513

Because a13b

4

51

81. Read a 1

81b

1/4

as “the fourth root of 1

81.”

Med

ioim

ages

/Pho

todi

sc/J

upite

r Im

ages

Caution

Intheexpressiona1/n,thereisnoreal-numbernthrootofawhennisevenanda , 0.Forexample,12642 1/2isnotarealnumber,becausethesquareofnorealnumberis264.

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A28 AppendixAAReviewofBasicAlgebra

summary of definitions of a1/n

d. 2321/5 5 2 1321/52 Read 321/5 as “the fifth root of 32.”

5 2 122 Because 25 5 32.

5 22

self Check 1 Simplify: a. 1001/2 b. 2431/5

now try exercise 15.

If n is even in the expression a1/n and the base contains variables, we often use absolute value symbols to guarantee that an even root is nonnegative.

149x22 1/2 5 7 0 x 0 Because 17 0 x 0 2 2 5 49x2. Since x could be negative, absolute value sym-

bols are necessary to guarantee that the square root is nonnegative.

116x42 1/4 5 2 0 x 0 Because 12 0 x 0 2 4 5 16x4. Since x could be negative, absolute value sym-

bols are necessary to guarantee that the fourth root is nonnegative.

1729x122 1/6 5 3x2 Because 13x22 6 5 729x12. Since x2 is always nonnegative, no absolute

value symbols are necessary. Read 1729x122 1/6 as “the sixth root of

729x12.”

If n is an odd number in the expression a1/n, the base a can be negative.

eXaMPLe 2 simplifying expressions with rational exponents

Simplify by using the definition of rational exponent.

a. 1282 1/3 5 22 Because 1222 3 5 28.

b. 123,1252 1/5 5 25 Because 1252 5 5 23,125.

c. a21

1,000b

1/3

5 2110

Because a21

10b

3

5 21

1,000.

self Check 2 Simplify: a. 121252 1/3 b. 12100,0002 1/5

now try exercise 19.

Ifnisoddintheexpressiona1/n,wedon’tneedtouseabsolutevaluesymbols,becauseoddrootscanbenegative.

1227x32 1/3 5 23x Because 123x2 3 5 227x3.

12128a72 1/7 5 22a Because 122a2 7 5 2128a7.

Wesummarizethedefinitionsconcerninga1/nasfollows.

If n is a natural number and a is a real number in the expression a1/n, then

If a $ 0, then a1/n is the nonnegative real number b such that bn 5 a.

If a , 0 eand n is odd, then a1/n is the real number b such that bn 5 a.and n is even, then a1/n is not a real number.

The following chart also shows the possibilities that can occur when simplify-ing a1/n.

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SectionA.3RationalExponentsandRadicalsA29

rule for rational exponents

a n a1/n Examples

a 5 0 n is a natural number.

01/n is the real number 0 because 0n 5 0.

01/2 5 0 because 02 5 0. 01/5 5 0 because 05 5 0.

a . 0 n is a natural number.

a1/n is the non-negative real number such that 1a1/n2 n 5 a.

161/2 5 4 because 42 5 16. 271/3 5 3 because 33 5 27.

a , 0 n is an odd natural number.

a1/n is the real number such that 1a1/n2 n 5 a.

12322 1/5 5 22 because 1222 5 5 232. 121252 1/3 5 25 because 1252 3 5 2125.

a , 0 n is an even natural number.

a1/n is not a real number.

1292 1/2 is not a real number. 12812 1/4 is not a real number.

2. define rational exponents Whose rational exponents are not 1

The definition of a1/n can be extended to include rational exponents whose numera-tors are not 1. For example, 43/2 can be written as either

141/22 3 or 1432 1/2 Because of the Power Rule, 1xm2 n 5 xmn.

This suggests the following rule.

If m and n are positive integers, the fraction mn is in lowest terms, and a1/n is a real number, then

am/n 5 1a1/n2m 5 1am2 1/n

In the previous rule, we can view the expression am/n in two ways:

1. 1a1/n2m: the mth power of the nth root of a 2. 1am2 1/n: the nth root of the mth power of a

For example, 12272 2/3 can be simplified in two ways:

12272 2/3 5 3 12272 1/3 42 or 12272 2/3 5 3 12272 2 41/3

5 1232 2 5 17292 1/3

5 9 5 9

As this example suggests, it is usually easier to take the root of the base first to avoid large numbers.

strategy for simplifying expressions of the Form a1/n

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A30 Appendix A A Review of Basic Algebra

Negative Rational Exponents

It is helpful to think of the phrase power over root when we see a rational exponent. The numerator of the fraction represents the power and the denominator represents the root. Begin with the root when simplifying to avoid large numbers.

If m and n are positive integers, the fraction mn is in lowest terms and a1/n is a

real number, then

a2m/n 51

am/n and 1

a2m/n 5 am/n 1a 2 02

EXAMPLE 3 Simplifying Expressions with Rational Exponents

We will apply the rules for rational exponents.

a. 253/2 5 1251/22 3 b. a2x6

1,000b

2/3

5 c a2x6

1,000b

1/3

d2

5 53 5 a2x2

10b

2

5 125 5x4

100

c. 3222/5 51

322/5 d. 1

8123/4 5 813/4

51

1321/52 2 5 1811/42 3

5122 5 33

514

5 27

Self Check 3 Simplify: a. 493/2 b. 1623/4 c. 1

127x3222/3

Now Try Exercise 43.

Because of the definition, rational exponents follow the same rules as integer exponents.

EXAMPLE 4 Using Exponent Rules to Simplify Expressions with Rational Exponents

Simplify each expression. Assume that all variables represent positive numbers, and write answers without using negative exponents.

a. 136x2 1/2 5 361/2x1/2 b. 1a1/3b2/32 6

1y32 2 5a6/3b12/3

y6

5 6x1/2 5a2b4

y6

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nth Root of a Nonnegative Number

Section A.3 Rational Exponents and Radicals A31

Definition of "n a

c. ax/2ax/4

ax/6 5 ax/21x/42x/6 d. c2c22/5

c4/5 d5/3

5 12c22/524/52 5/3

5 a6x/1213x/1222x/12 5 3 1212 1c26/52 45/3

5 a7x/12 5 1212 5/3 1c26/52 5/3

5 21c230/15

5 2c22

5 21c2

Self Check 4 UsethedirectionsforExample4:

a. a y2

49b

1/2

b.b3/7b2/7

b4/7 c.19r2s2 1/2

rs23/2

Now Try Exercise 59.

3. Define Radical ExpressionsRadicalsignscanalsobeusedtoexpressrootsofnumbers.

If nisanaturalnumbergreaterthan1andif a1/nisarealnumber,then

"na 5 a1/n

Intheradicalexpression"na,thesymbol"istheradicalsign,aistheradicand,

andnistheindex(ortheorder)oftheradicalexpression.If theorderis2,theex-

pressionisasquareroot,andwedonotwritetheindex.

"a 5 "2 a

If theindexofaradicalis3,wecalltheradicalacuberoot.

If nisanaturalnumbergreaterthan1anda $ 0,then"naisthenonnegative

numberwhosenthpowerisa.

Q"naRn

5 a

In the expression "na, there is no real-number nth root of a when n is even and a , 0. For

example, "264 is not a real number, because the square of no real number is 264.

If 2 is substituted for n in the equation Q"naRn

5 a, we have

Q"2 aR25 Q"aR2

5 "a"a 5 a for a $ 0

This shows that if a number a can be factored into two equal factors, either of those factors is a square root of a. Furthermore, if a can be factored into n equal factors, any one of those factors is an nth root of a.

If n is an odd number greater than 1 in the expression "na, the radicand can be negative.

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A32 AppendixAAReviewofBasicAlgebra

strategy for simplifying expressions of the Form "n a

summary of definitions of "n a

eXaMPLe 5 Finding nth roots of real numbers

We apply the definitions of cube root and fifth root.

a. "3227 5 23 Because 1232 3 5 227.

b. "328 5 22 Because 1222 3 5 28.

c. Å3

227

1,0005 2

310

Because a23

10b

3

5 227

1,000.

d. 2"52243 5 2Q"5

2243R 5 2 1232 5 3

self Check 5 Find each root: a. "3 216 b. Å5

2132

now try exercise 69.

We summarize the definitions concerning "na as follows.

If n is a natural number greater than 1 and a is a real number, then

If a $ 0, then "na is the nonnegative real number such that Q"n

aRn5 a.

If a , 0 eand n is odd, then "na is the real number such that Q"n

aRn5 a.

and n is even, then "na is not a real number.

The following chart also shows the possibilities that can occur when simplify-

ing "na.

a n "n a Examples

a 5 0 n is a natural

number

greater than

1.

"n0 is the real

number 0

because 0n 5 0.

"3 0 5 0 because 03 5 0.

"5 0 5 0 because 05 5 0.

a . 0 n is a natural

number

greater than

1.

"na is the non-

negative real number such

that Q"naRn

5 a.

"16 5 4 because 42 5 16.

"3 27 5 3 because 33 5 27.

a , 0 n is an odd

natural num-

ber greater

than 1.

"na is the real

number such

that Q"naRn

5 a.

"5232 5 22 because 1222 5 5 232.

"32125 5 25 because 1252 3 5 2125.

a , 0 n is an even

natural

number.

"na is not a real

number.

"29 is not a real number.

"4281 is not a real number.

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SectionA.3RationalExponentsandRadicalsA33

Multiplication and division Properties of radicals

We have seen that if a1/n is real, then am/n 5 1a1/n2m 5 1am2 1/n.This same fact can be stated in radical notation.

am/n 5 1"na2m 5 "n

am

Thus, the mth power of the nth root of a is the same as the nth root of the mth power

of a. For example, to find "3 272, we can proceed in either of two ways:

"3 272 5 Q"3 27R25 32 5 9 or "3 272 5 "3 729 5 9

By definition, "a2 represents a nonnegative number. If a could be negative, we

must use absolute value symbols to guarantee that "a2 will be nonnegative. Thus,

if a is unrestricted,

"a2 5 0 a 0A similar argument holds when the index is any even natural number. The sym-

bol "4 a4, for example, means the positive fourth root of a4. Thus, if a is unrestricted,

"4 a4 5 0 a 0

eXaMPLe 6 simplifying radical expressions

If x is unrestricted, simplify a. "6 64x6 b. "3 x3 c. "9x8

sOLUtiOn We apply the definitions of sixth roots, cube roots, and square roots.

a. "6 64x6 5 2 0 x 0 Use absolute value symbols to guarantee that the result will be nonnegative.

b. "3 x3 5 x Because the index is odd, no absolute value symbols are needed.

c. "9x8 5 3x4 Because 3x4 is always nonnegative, no absolute value symbols are needed.

self Check 6 Use the directions for Example 6:

a. "4 16x4 b. "3 27y3 c. "4 x8

now try exercise 73.

4. simplify and Combine radicalsMany properties of exponents have counterparts in radical notation. For example,

since a1/nb1/n 5 1ab2 1/n and a1/n

b1/n 5 1ab2 1/n and 1b 2 02 , we have the following.

If all expressions represent real numbers,

"na"n

b 5 "nab

"na

"nb

5 Ån a

b 1b 2 02

In words, we say

The product of two nth roots is equal to the nth root of their product.

The quotient of two nth roots is equal to the nth root of their quotient.Not For Sale

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A34 AppendixAAReviewofBasicAlgebra

Caution Thesepropertiesinvolvethenthrootoftheproductoftwonumbersorthenthrootofthequo-tientoftwonumbers.Thereisnosuchpropertyforsumsordifferences.Forexample,

"9 1 4 2 "9 1 "4,because

"9 1 4 5 "13 but "9 1 "4 5 3 1 2 5 5

and"13 2 5.Ingeneral,

"a 1 b 5 "a 1 "b and "a 2 b 5 "a 2 "b

Numbers that are squares of positive integers, such as

1, 4, 9, 16, 25, and 36

are called perfectsquares. Expressions such as 4x2 and 19x6 are also perfect squares,

because each one is the square of another expression with integer exponents and rational coefficients.

4x2 5 12x2 2 and 19

x6 5 a13

x3b2

Numbers that are cubes of positive integers, such as

1, 8, 27, 64, 125, and 216

are called perfectcubes. Expressions such as 64x3 and 127x9 are also perfect cubes,

because each one is the cube of another expression with integer exponents and ratio-nal coefficients.

64x3 5 14x2 3 and 127

x9 5 a13

x3b3

There are also perfect fourth powers, perfect fifth powers, and so on.We can use perfect powers and the Multiplication Property of Radicals to sim-

plify many radical expressions. For example, to simplify "12x5, we factor 12x5 so that one factor is the largest perfect square that divides 12x5. In this case, it is 4x4. We then rewrite 12x5 as 4x4 ? 3x and simplify.

"12x5 5 "4x4 ? 3x Factor 12x5 as 4x4 ? 3x.

5 "4x4"3x Use the Multiplication Property of Radicals: "ab 5 "a"b.

5 2x2"3x "4x4 5 2x2

To simplify "3 432x9y, we find the largest perfect-cube factor of 432x9y (which is 216x9) and proceed as follows:

"3 432x9y 5 "3 216x9 ? 2y Factor 432x9y as 216x9 ? 2y.

5 "3 216x9 "3 2y Use the Multiplication Property of Radicals: "3 ab 5 "3 a"3 b.

5 6x3"3 2y "3 216x9 5 6x3

Radical expressions with the same index and the same radicand are called like

or similarradicals. We can combine the like radicals in 3"2 1 2"2 by using the

Distributive Property.

3"2 1 2"2 5 13 1 22"2

5 5"2

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SectionA.3RationalExponentsandRadicalsA35

This example suggests that to combine like radicals, we add their numerical coefficients and keep the same radical.

When radicals have the same index but different radicands, we can often change

them to equivalent forms having the same radicand. We can then combine them. For

example, to simplify "27 2 "12, we simplify both radicals and combine like radicals.

"27 2 "12 5 "9 ? 3 2 "4 ? 3 Factor 27 and 12.

5 "9"3 2 "4"3 "ab 5 "a"b

5 3"3 2 2"3 "9 5 3 and "4 5 2.

5 "3 Combine like radicals.

eXaMPLe 7 adding and subtracting radical expressions

Simplify: a. "50 1 "200 b. 3z"5 64z 2 2"5 2z6

sOLUtiOn We will simplify each radical expression and then combine like radicals.

a. "50 1 "200 5 "25 ? 2 1 "100 ? 2

5 "25"2 1 "100"2

5 5"2 1 10"2

5 15"2

b. 3z"5 64z 2 2"5 2z6 5 3z"5 32 ? 2z 2 2"5 z5 ? 2z

5 3z"5 32 "5 2z 2 2"5 z5"5 2z

5 3z 122"5 2z 2 2z"5 2z

5 6z"5 2z 2 2z"5 2z

5 4z"5 2z

self Check 7 Simplify: a. "18 2 "8 b. 2"3 81a4 1 a"3 24a

now try exercise 85.

5. rationalize denominators and numeratorsBy rationalizingthedenominator, we can write a fraction such as

"5

"3

as a fraction with a rational number in the denominator. All that we must do is multiply both the numerator and the denominator by "3. (Note that "3"3 is the rational number 3.)

"5

"35

"5"3

"3"35

"153

To rationalize the numerator, we multiply both the numerator and the denomi-nator by "5. (Note that "5"5 is the rational number 5.)

"5

"35

"5"5

"3"55

5

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A36 AppendixAAReviewofBasicAlgebra

eXaMPLe 8 rationalizing the denominator of a radical expression

Rationalize each denominator and simplify. Assume that all variables represent positive numbers.

a. 1

"7 b. Å

3 34

c. Å3x

d. Å3a3

5x5

sOLUtiOn We will multiply both the numerator and the denominator by a radical that will make the denominator a rational number.

a. 1

"75

1"7

"7"7 b. Å

3 34

5"3 3

"3 4

5"7

7 5

"3 3"3 2

"3 4"3 2

Multiply numerator and

denominator by "3 2, because

"3 4"3 2 5 "3 8 5 2.

5"3 6

"3 8

5"3 6

2

c. Å3x

5"3

"x d. Å

3a3

5x5 5"3a3

"5x5

5"3"x

"x"x 5

"3a3"5x

"5x5"5x

Multiply numerator and denominator by "5x, because

"5x5 "5x 5 "25x6 5 5x3.

5"3x

x 5

"15a3x

"25x6

5"a2"15ax

5x3

5a"15ax

5x3

self Check 8 Use the directions for Example 8:

a. 6

"6 b. Å

3 25x

now try exercise 101.

eXaMPLe 9 rationalizing the numerator of a radical expression

Rationalize each numerator and simplify. Assume that all variables represent posi-

tive numbers: a. "x

7 b.

2"3 9x3

sOLUtiOn We will multiply both the numerator and the denominator by a radical that will make the numerator a rational number.

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SectionA.3RationalExponentsandRadicalsA37

a. "x

75

"x ? "x

7"x b.

2"3 9x3

52"3 9x ? "3 3x2

3"3 3x2

5x

7"x 5

2"3 27x3

3"3 3x2

52 13x2

3"3 3x2

52x

"3 3x2 Divide out the 3’s.

self Check 9 Use the directions for Example 9:

a. "2x

5 b.

3"3 2y2

6

now try exercise 111.

After rationalizing denominators, we often can simplify an expression.

eXaMPLe 10 rationalizing denominators and simplifying

Simplify: Å12

1 Å18

.

sOLUtiOn We will rationalize the denominators of each radical and then combine like radicals.

Å12

1 Å18

51

"21

1

"8 Å

12

5"1

"25

1

"2; Å

18

5"1

"85

1

"8

51"2

"2"21

1"2

"8"2

5"2

21

"2

"16

5"2

21

"24

53"2

4

self Check 10 Simplify: Å3 x

22 Å

3 x16

.

now try exercise 115.

Another property of radicals can be derived from the properties of exponents. If all of the expressions represent real numbers,

#n "m x 5 "nx1/m 5 1x1/m2 1/n 5 x1/ 1mn2 5

mn"x

#"x 5m "x1/n 5 1x1/n2 1/m 5 x1/1nm2 5

mn"x

These results are summarized in the following theorem (a fact that can be proved).

m n

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A38 AppendixAAReviewofBasicAlgebra

theorem If all of the expressions involved represent real numbers, then

m#"n

x 5 #n "m x 5mn"x

We can use the previous theorem to simplify many radicals. For example,

#3 "8 5 #"3 8 5 "2

Rational exponents can be used to simplify many radical expressions, as shown in the following example.

eXaMPLe 11 simplifying radicals Using the Previously stated theorem

Simplify. Assume that x and y are positive numbers.

a. "6 4 b. 12"x3 c. "9 8y3

sOLUtiOn In each case, we will write the radical as an exponential expression, simplify the resulting expression, and write the final result as a radical.

a. "6 4 5 41/6 5 1222 1/6 5 22/6 5 21/3 5 "3 2

b.12"x3 5 x3/12 5 x1/4 5 "4 x

c. "9 8y3 5 123y32 1/9 5 12y2 3/9 5 12y2 1/3 5 "3 2y

self Check 11 Simplify: a. "4 4 b. "9 27x3

now try exercise 117.

self Check answers 1. a. 10 b. 3 2. a. 25 b. 210 3. a. 343 b. 18

c. 9x2 4. a. y7

b. b1/7 c. 3s2 5. a. 6 b. 212

6. a. 2 0 x 0

b. 3y c. x2 7. a. "2 b. 8a"3 3a 8. a. "6 b. "3 50x2

5x

9. a. 2x

5"2x b.

y

"3 4y 10.

"3 4x4

11. a. "2 b. "3 3x

Exercises A.3Getting ready You should be able to complete these vocabulary and concept statements before you proceed to the practice exercises.

Fill in the blanks. 1. If a 5 0 and n is a natural number, then a1/n 5 .

2. If a . 0 and n is a natural number, then a1/n is a number.

3. If a , 0 and n is an even number, then a1/n is a real number.

4. 62/3 can be written as or .

5. "na 5 6. "a2 5

7. "na"n

b 5 8. Ån a

b5

9. "x 1 y "x 1 "y

10. m#"n

x or # m"nx can be written as .

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SectionA.3RationalExponentsandRadicalsA39

Practice Simplify each expression.

11. 91/2 12. 81/3

13. a 125

b1/2

14. a 16625

b1/4

15. 2811/4 16. 2a 827

b1/3

17. 110,0002 1/4 18. 1,0241/5

19. a2278b

1/3

20. 2641/3

21. 12642 1/2 22. 121252 1/3

Simplify each expression. Use absolute value symbols when necessary.

23. 116a22 1/2 24. 125a42 1/2

25. 116a42 1/4 26. 1264a32 1/3

27. 1232a52 1/5 28. 164a62 1/6

29. 12216b62 1/3 30. 1256t82 1/4

31. a16a4

25b2b1/2

32. a2a5

32b10b1/5

33. a21,000x6

27y3 b1/3

34. a 49t2

100z4b1/2

Simplify each expression. Write all answers without using negative exponents.

35. 43/2 36. 82/3

37. 2163/2 38. 1282 2/3

39. 21,0002/3 40. 1003/2

41. 6421/2 42. 2521/2

43. 6423/2 44. 4923/2

45. 2923/2 46. 1227222/3

47. a49b

5/2

48. a2581

b3/2

49. a22764

b22/3

50. a1258

b24/3

Simplify each expression. Assume that all variables rep-resent positive numbers. Write all answers without using negative exponents.

51. 1100s42 1/2 52. 164u6v32 1/3

53. 132y10z5221/5 54. 1625a4b8221/4

55. 1x10y52 3/5 56. 164a6b122 5/6

57. 1r8s16223/4 58. 128x9y12222/3

59. a28a6

125b9b2/3

60. a 16x4

625y8b3/4

61. a 27r6

1,000s12b22/3

62. a232m10

243n15b22/5

63.a2/5a4/5

a1/5 64. x6/7x3/7

x2/7x5/7

Simplify each radical expression.

65. "49 66. "81

67. "3 125 68. "3264

69. "32125 70. "5

2243

71. Å5

232

100,000 72. Å

4 256625

Simplify each expression, using absolute value symbols when necessary. Write answers without using negative exponents.

73. "36x2 74. 2"25y2

75. "9y4 76. "a4b8

77. "3 8y3 78. "3227z9

79. Å4 x4y8

z12 80. Å5 a10b5

c15

Simplify each expression. Assume that all variables represent positive numbers so that no absolute value symbols are needed. 81. "8 2 "2 82. "75 2 2"27

83."200x2 1 "98x2 84. "128a3 2 a"162a

85. 2"48y5 2 3y"12y3 86. y"112y 1 4"175y3

87. 2"3 81 1 3"3 24 88. 3"4 32 2 2"4 162

89."4 768z5 1 "4 48z5 90. 22"5 64y2 1 3"5 486y2

91."8x2y 2 x"2y 1 "50x2y

92. 3x"18x 1 2"2x3 2 "72x3

93. "3 16xy4 1 y"3 2xy 2 "3 54xy4

94. "4 512x5 2 "4 32x5 1 "4 1,250x5

Rationalize each denominator and simplify. Assume that all variables represent positive numbers.

95. 3

"3 96.

6

"5

97. 2

"x 98.

8

"y

99. 2

"3 2 100.

4d

"3 9

101. 5a

"3 25a 102.

7

"3 36c

103. 2b

"4 3a2 104. Å

x2y

105. Å3 2u4

9v 106. Å

32

3s5

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A40 AppendixAAReviewofBasicAlgebra

Rationalize each numerator and simplify. Assume that all variables are positive numbers.

107. "510

108. "y

3

109. "3 9

3 110.

"3 16b2

16

111. "5 16b3

64a 112. Å

3x57

Rationalize each denominator and simplify.

113. Å13

2 Å127

114. Å3 1

21 Å

3 116

115.Åx8

2 Åx2

1 Åx32

116. Å3 y

41 Å

3 y32

2 Å3 y

500

Simplify each radical expression.

117. "4 9 118. "6 27

119.10"16x6 120. "6 27x9

discovery and WritingWe often can multiply and divide radicals with different indi-ces. For example, to multiply "3 by "3 5, we first write each radical as a sixth root

"3 5 31/2 5 33/6 5 "6 33 5 "6 27

"3 5 5 51/3 5 52/6 5 "6 52 5 "6 25

and then multiply the sixth roots.

"3"3 5 5 "6 27"6 25 5 "6 1272 1252 5 "6 675

Division is similar.

Use this idea to write each of the following expressions as a single radical.

121. "2"3 2 122. "3"3 5

123. "4 3

"2 124.

"3 2

"5

125. For what values of x does "4 x4 5 x? Explain.

126. If all of the radicals involved represent real numbers and y 2 0, explain why

Ån x

y5

"nx

"ny

127. If all of the radicals involved represent real numbers and there is no division by 0, explain why

axyb

2m/n

5 Ån ym

xm

128. The definition of xm/n requires that "nx be a real

number. Explain why this is important. (Hint:

Consider what happens when n is even, m is odd,

and x is negative.)

review 129. Write 22 , x # 5 using interval notation.

130. Write the expression 0 3 2 x 0 without using absolute value symbols. Assume that x . 4.

Evaluate each expression when x 5 22 and y 5 3.

131. x2 2 y2 132. xy 1 4y

x

133. Write 617,000,000 in scientific notation.

134. Write 0.00235 3 104 in standard notation.

A.4 PolynomialsInthissection,wewilllearnto

1.Definepolynomials.2.Addandsubtractpolynomials.3.Multiplypolynomials.4.Rationalizedenominators.5.Dividepolynomials.

Football is one of the most popular sports in the United States. Brett Favre, one of the most talented quarterbacks ever to play the game, holds the record for the most career NFL touchdown passes, the most NFL pass completions, and the most passing yards. He led the Green Bay Packers to the Super Bowl and most recently played for the Minnesota Vikings. His nickname is Gunslinger.©

PCN

Pho

togr

aphy

/Ala

my

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SectionA.4PolynomialsA41

An algebraic expression can be used to model the trajectory or path of a foot-ball when passed by Brett Favre. Suppose the height in feet, t seconds after the football leaves Brett’s hand, is given by the algebraic expression

20.1t2 1 t 1 5.5

At a time of t 5 3 seconds, we see that the height of the football is

20.1 132 2 1 3 1 5.5 5 7.6 ft.

Algebraic expressions like 20.1t2 1 t 1 5.5 are called polynomials, and we will study them in this section.

1. define PolynomialsA monomial is a real number or the product of a real number and one or more variables with whole-number exponents. The number is called the coefficient of the variables. Some examples of monomials are

3x, 7ab2, 25ab2c4, x3, and 212

with coefficients of 3, 7, 25, 1, and 212, respectively.The degree of a monomial is the sum of the exponents of its variables. All non-

zero constants (except 0) have a degree of 0.

The degree of 3x is 1. The degree of 7ab2 is 3.

The degree of 25ab2c4 is 7. The degree of x3 is 3.

The degree of 212 is 0 (since 212 5 212x0). 0 has no defined degree.

A monomial or a sum of monomials is called a polynomial. Each monomial in that sum is called a term of the polynomial. A polynomial with two terms is called a binomial, and a polynomial with three terms is called a trinomial.

Monomials Binomials Trinomials

3x2 2a 1 3b x2 1 7x 2 4

225xy 4x3 2 3x2 4y4 2 2y 1 12

a2b3c 22x3 2 4y2 12x3y2 2 8xy 2 24

The degreeofapolynomial is the degree of the term in the polynomial with highest degree. The only polynomial with no defined degree is 0, which is called the zeropolynomial. Here are some examples.

• 3x2y3 1 5xy2 1 7 is a trinomial of 5th degree, because its term with highest degree (the first term) is 5.

• 3ab 1 5a2b is a binomial of degree 3.

• 5x 1 3y2 1 "4 3z4 2 "7 is a polynomial, because its variables have whole-number exponents. It is of degree 4.

• 27y1/2 1 3y2 1 "5 3z is not a polynomial, because one of its variables (y in the first term) does not have a whole-number exponent.

If two terms of a polynomial have the same variables with the same exponents, they are like or similar terms. To combine the like terms in the sum 3x2y 1 5x2y or the difference 7xy2 2 2xy2, we use the Distributive Property:

3x2y 1 5x2y 5 13 1 52x2y 7xy2 2 2xy2 5 17 2 22xy2

5 8x2y 5 5xy2

This illustrates that to combine like terms, we add (or subtract) their coefficients and keep the same variables and the same exponents.Not For Sale

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A42 AppendixAAReviewofBasicAlgebra

2. add and subtract PolynomialsRecall that we can use the Distributive Property to remove parentheses enclosing the terms of a polynomial. When the sign preceding the parentheses is +, we simply drop the parentheses:

1 1a 1 b 2 c2 5 11 1a 1 b 2 c2 5 1a 1 1b 2 1c

5 a 1 b 2 c

When the sign preceding the parentheses is 2, we drop the parentheses and the 2 sign and change the sign of each term within the parentheses.

2 1a 1 b 2 c2 5 21 1a 1 b 2 c2 5 21a 1 1212b 2 1212c 5 2a 2 b 1 c

We can use these facts to add and subtract polynomials. To add (or subtract) polynomials, we remove parentheses (if necessary) and combine like terms.

eXaMPLe 1 adding Polynomials

Add: 13x3y 1 5x2 2 2y2 1 12x3y 2 5x2 1 3x2 . sOLUtiOn To add the polynomials, we remove parentheses and combine like terms.

13x3y 1 5x2 2 2y2 1 12x3y 2 5x2 1 3x2 5 3x3y 1 5x2 2 2y 1 2x3y 2 5x2 1 3x

5 3x3y 1 2x3y 1 5x2 2 5x2 2 2y 1 3x Use the Commutative Property to

rearrange terms.

5 5x3y 2 2y 1 3x Combine like terms.

We can add the polynomials in a vertical format by writing like terms in a col-umn and adding the like terms, column by column.

3x3y 1 5x2 2 2y2x3y 2 5x2 1 3x5x3y 2 2y 1 3x

self Check 1 Add: 14x2 1 3x 2 52 1 13x2 2 5x 1 72 . now try exercise 21.

eXaMPLe 2 subtracting Polynomials

Subtract: 12x2 1 3y22 2 1x2 2 2y2 1 72 . sOLUtiOn To subtract the polynomials, we remove parentheses and combine like terms.

12x2 1 3y22 2 1x2 2 2y2 1 72 5 2x2 1 3y2 2 x2 1 2y2 2 7

5 2x2 2 x2 1 3y2 1 2y2 2 7 Use the Commutative Property to

rearrange terms.

5 x2 1 5y2 2 7 Combine like terms.

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SectionA.4PolynomialsA43

We can subtract the polynomials in a vertical format by writing like terms in a column and subtracting the like terms, column by column.

2x2 1 3y2 2x2 2 x2 5 12 2 12x2

2 1x2 2 2y2 1 72 3y2 2 1222y2 5 3y2 1 2y2 5 13 1 22y2

x2 1 5y2 2 7

self Check 2 Subtract: 14x2 1 3x 2 52 2 13x2 2 5x 1 72 . now try exercise 23.

We can also use the Distributive Property to remove parentheses enclosing sev-eral terms that are multiplied by a constant. For example,

4 13x2 2 2x 1 62 5 4 13x22 2 4 12x2 1 4 162 5 12x2 2 8x 1 24

This example suggests that to add multiples of one polynomial to another, or to subtract multiples of one polynomial from another, we remove parentheses and com-bine like terms.

eXaMPLe 3 Using the distributive Property and Combining Like terms

Simplify: 7x 12y2 1 13x22 2 5 1xy2 2 13x32 . sOLUtiOn 7x 12y2 1 13x22 2 5 1xy2 2 13x32 5 14xy2 1 91x3 2 5xy2 1 65x3 Use the Distributive Property to remove parentheses.

5 14xy2 2 5xy2 1 91x3 1 65x3 Use the Commutative Property to rearrange terms.

5 9xy2 1 156x3 Combine like terms.

self Check 3 Simplify: 3 12b2 2 3a2b2 1 2b 1b 1 a22 . now try exercise 30.

3. Multiply PolynomialsTo find the product of 3x2y3z and 5xyz2, we proceed as follows:

13x2y3z2 15xyz22 5 3 ? x2 ? y3 ? z ? 5 ? x ? y ? z2

5 3 ? 5 ? x2 ? x ? y3 ? y ? z ? z2 Use the Commutative Property to

rearrange terms.

5 15x3y4z3

This illustrates that to multiply two monomials, we multiply the coefficients and then multiply the variables.

To find the product of a monomial and a polynomial, we use the Distributive Property.

3xy2 12xy 1 x2 2 7yz2 5 3xy2 12xy2 1 13xy22 1x22 2 13xy22 17yz2 5 6x2y3 1 3x3y2 2 21xy3z

This illustrates that to multiply a polynomial by a monomial, we multiply each term of the polynomial by the monomial.

0 2 7 5 27

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A44 AppendixAAReviewofBasicAlgebra

special Product Formulas

To multiply one binomial by another, we use the Distributive Property twice.

eXaMPLe 4 Multiplying Binomials

Multiply: a. 1x 1 y2 1x 1 y2 b. 1x 2 y2 1x 2 y2 c. 1x 1 y2 1x 2 y2

sOLUtiOn a. 1x 1 y2 1x 1 y2 5 1x 1 y2x 1 1x 1 y2y 5 x2 1 xy 1 xy 1 y2

5 x2 1 2xy 1 y2

b. 1x 2 y2 1x 2 y2 5 1x 2 y2x 2 1x 2 y2y 5 x2 2 xy 2 xy 1 y2

5 x2 2 2xy 1 y2

c. 1x 1 y2 1x 2 y2 5 1x 1 y2x 2 1x 1 y2y 5 x2 1 xy 2 xy 2 y2

5 x2 2 y2

self Check 4 Multiply: a. 1x 1 22 1x 1 22 b. 1x 2 32 1x 2 32 c. 1x 1 42 1x 2 42

now try exercise 45.

The products in Example 4 are called specialproducts. Because they occur so often, it is worthwhile to learn their forms.

1x 1 y2 2 5 1x 1 y2 1x 1 y2 5 x2 1 2xy 1 y2

1x 2 y2 2 5 1x 2 y2 1x 2 y2 5 x2 2 2xy 1 y2

1x 1 y2 1x 2 y2 5 x2 2 y2

Rememberthat 1x 1 y2 2and 1x 2 y2 2havetrinomialsfortheirproductsandthat

1x 1 y2 2 2 x2 1 y2 and 1x 2 y2 2 2 x2 2 y2

Forexample,

13 1 52 2 2 32 1 52 and 13 2 52 2 2 32 2 52

82 2 9 1 25 1222 2 2 9 2 25

64 2 34 4 2 216

We can use the FOILmethod to multiply one binomial by another. FOIL is an acronym for First terms, Outer terms, Inner terms, and Last terms. To use this method to multiply 3x 2 4 by 2x 1 5, we write

Caution

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SectionA.4PolynomialsA45

First terms Last terms

13x 2 42 12x 1 52 5 3x 12x2 1 3x 152 2 4 12x2 2 4 152 5 6x2 1 15x 2 8x 2 20

5 6x2 1 7x 2 20

In this example,

• the product of the first terms is 6x2,• the product of the outer terms is 15x,• the product of the inner terms is 28x, and• the product of the last terms is 220.

The resulting like terms of the product are then combined.

eXaMPLe 5 Using the FOiL Method to Multiply Polynomials

Use the FOIL method to multiply: Q"3 1 xRQ2 2 "3xR.

sOLUtiOn 1"3 1 x2 12 2 "3x2 5 2"3 2 "3"3x 1 2x 2 x"3x

5 2"3 2 3x 1 2x 2 "3x2

5 2"3 2 x 2 "3x2

self Check 5 Multiply: Q2x 1 "3RQx 2 "3R. now try exercise 63.

To multiply a polynomial with more than two terms by another polynomial, we multiply each term of one polynomial by each term of the other polynomial and combine like terms whenever possible.

eXaMPLe 6 Multiplying Polynomials

Multiply: a. 1x 1 y2 1x2 2 xy 1 y22 b. 1x 1 32 3

sOLUtiOn a. 1x 1 y2 1x2 2 xy 1 y22 5 x3 2 x2y 1 xy2 1 yx2 2 xy2 1 y3

5 x3 1 y3

b. 1x 1 32 3 5 1x 1 32 1x 1 32 2

5 1x 1 32 1x2 1 6x 1 92 5 x3 1 6x2 1 9x 1 3x2 1 18x 1 27

5 x3 1 9x2 1 27x 1 27

self Check 6 Multiply: 1x 1 22 12x2 1 3x 2 12 . now try exercise 67.

We can use a vertical format to multiply two polynomials, such as the polyno-mials given in Self Check 6. We first write the polynomials as follows and draw a line beneath them. We then multiply each term of the upper polynomial by each term of the lower polynomial and write the results so that like terms appear in each column. Finally, we combine like terms column by column.

Outer terms

Inner terms

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A46 AppendixAAReviewofBasicAlgebra

Conjugate Binomials

2x2 1 3x 2 1x 1 2

4x2 1 6x 2 2 Multiply 2x2 1 3x 2 1 by 2.

2x3 1 3x2 2 x Multiply 2x2 1 3x 2 1 by x.

2x3 1 7x2 1 5x 2 2 In each column, combine like terms.

If n is a whole number, the expressions an 1 1 and 2an 2 3 are polynomials and we can multiply them as follows:

1an 1 12 12an 2 32 5 2a2n 2 3an 1 2an 2 3

5 2a2n 2 an 2 3 Combine like terms.

We can also use the methods previously discussed to multiply expressions that are not polynomials, such as x22 1 y and x2 2 y21.

1x22 1 y2 1x2 2 y212 5 x2212 2 x22y21 1 x2y 2 y121

5 x0 21

x2y1 x2y 2 y0

5 1 21

x2y1 x2y 2 1 x0 5 1 and y0 5 1.

5 x2y 21

x2y

4. rationalize denominatorsIf the denominator of a fraction is a binomial containing square roots, we can use the product formula 1x 1 y2 1x 2 y2 to rationalize the denominator. For example, to rationalize the denominator of

6

"7 1 2

To rationalize a denominator means to change

the denominator into a rational number.

we multiply the numerator and denominator by "7 2 2 and simplify.

6

"7 1 25

6 1"7 2 221"7 1 22 1"7 2 22

"7 2 2

"7 2 25 1

56 1"7 2 22

7 2 4

56 1"7 2 22

3 Here the denominator is a rational number.

5 2 1"7 2 22In this example, we multiplied both the numerator and the denominator of the

given fraction by "7 2 2. This binomial is the same as the denominator of the

given fraction "7 1 2, except for the sign between the terms. Such binomials are

called conjugatebinomials or radicalconjugates.

Conjugatebinomials are binomials that are the same except for the sign between their terms. The conjugate of a 1 b is a 2 b, and the conjugate of a 2 b is a 1 b.

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SectionA.4PolynomialsA47

eXaMPLe 7 rationalizing the denominator of a radical expression

Rationalize the denominator: "3x 2 "2

"3x 1 "2 1x . 02 .

sOLUtiOn We multiply the numerator and the denominator by "3x 2 "2 (the conjugate of

"3x 1 "22 and simplify.

"3x 2 "2

"3x 1 "25

Q"3x 2 "2RQ"3x 2 "2RQ"3x 1 "2RQ"3x 2 "2R

"3x 2 "2

"3x 2 "25 1

5"3x"3x 2 "3x"2 2 "2"3x 1 "2"2

Q"3xR22 Q"2R2

53x 2 "6x 2 "6x 1 2

3x 2 2

53x 2 2"6x 1 2

3x 2 2

self Check 7 Rationalize the denominator: "x 1 2

"x 2 2.

now try exercise 89.

In calculus, we often rationalize a numerator.

eXaMPLe 8 rationalizing the numerator of a radical expression

Rationalize the numerator: "x 1 h 2 "x

h.

sOLUtiOn To rid the numerator of radicals, we multiply the numerator and the denominator by the conjugate of the numerator and simplify.

"x 1 h 2 "x

h5

Q"x 1 h 2 "xRQ"x 1 h 1 "xRhQ"x 1 h 1 "xR

"x 1 h 1 "x

"x 1 h 1 "x5 1

5x 1 h 2 x

hQ"x 1 h 1 "xR Here the numerator has no radicals.

5h

hQ"x 1 h 1 "xR

51

"x 1 h 1 "x Divide out the common factor of h.

self Check 8 Rationalize the numerator: "4 1 h 2 2

h.

now try exercise 99.Not For Sale

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A48 AppendixAAReviewofBasicAlgebra

5. divide PolynomialsTo divide monomials, we write the quotient as a fraction and simplify by using the rules of exponents. For example,

6x2y3

22x3y5 23x223y321

5 23x21y2

5 23y2

xTo divide a polynomial by a monomial, we write the quotient as a fraction,

write the fraction as a sum of separate fractions, and simplify each one. For example, to divide 8x5y4 1 12x2y5 2 16x2y3 by 4x3y4, we proceed as follows:

8x5y4 1 12x2y5 2 16x2y3

4x3y4 58x5y4

4x3y4 112x2y5

4x3y4 1216x2y3

4x3y4

5 2x2 13yx

24

xy

To divide two polynomials, we can use long division. To illustrate, we consider the division

2x2 1 11x 2 30x 1 7

which can be written in long division form as

x 1 7q2x2 1 11x 2 30

The binomial x 1 7 is called the divisor, and the trinomial 2x2 1 11x 2 30 is called the dividend. The final answer, called the quotient, will appear above the long divi-sion symbol.

We begin the division by asking “What monomial, when multiplied by x, gives 2x2?” Because x ? 2x 5 2x2, the answer is 2x. We place 2x in the quotient, multiply each term of the divisor by 2x, subtract, and bring down the –30.

2xx 1 7q2x2 1 11x 2 30

2x2 1 14x2 3x 2 30

We continue the division by asking “What monomial, when multiplied by x, gives 23x?” We place the answer, 23, in the quotient, multiply each term of the divisor by 23, and subtract. This time, there is no number to bring down.

2x 2 3x 1 7q2x2 1 11x 2 30

2x2 1 14x2 3x 2 302 3x 2 21

29

Because the degree of the remainder, 29, is less than the degree of the divisor, the division process stops, and we can express the result in the form

quotient 1remainder

divisor

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SectionA.4PolynomialsA49

Thus,

2x2 1 11x 2 30x 1 7

5 2x 2 3 129

x 1 7

eXaMPLe 9 Using Long division to divide Polynomials

Divide 6x3 2 11 by 2x 1 2.

sOLUtiOn We set up the division, leaving spaces for the missing powers of x in the dividend.

2x 1 2q6x3 2 11

The division process continues as usual, with the following results:

3x2 2 3x 1 32x 1 2q6x3 2 11

6x3 1 6x2

2 6x2

2 6x2 2 6x1 6x 2 111 6x 1 6

2 17

Thus, 6x3 2 112x 1 2

5 3x2 2 3x 1 3 1217

2x 1 2.

self Check 9 Divide: 3x 1 1q9x2 2 1.

now try exercise 113.

eXaMPLe 10 Using Long division to divide Polynomials

Divide 23x3 2 3 1 x5 1 4x2 2 x4 by x2 2 3.

sOLUtiOn The division process works best when the terms in the divisor and dividend are writ-ten with their exponents in descending order.

x3 2 x2 1 1x2 2 3qx5 2 x4 2 3x3 1 4x2 2 3

x5 2 3x3

2 x4 1 4x2

2 x4 1 3x2

x2 2 3x2 2 3

0

Thus, 23x3 2 3 1 x5 1 4x2 2 x4

x2 2 35 x3 2 x2 1 1.

self Check 10 Divide: x2 1 1q3x2 2 x 1 1 2 2x3 1 3x4.

now try exercise 115.

Comment

InExample9,wecouldwritethemissingpowersofxusingcoefficientsof0.

2x 1 2q6x3 1 0x2 1 0x 2 11

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A50 AppendixAAReviewofBasicAlgebra

self Check answers 1. 7x2 2 2x 1 2 2. x2 1 8x 2 12 3. 8b2 2 7a2b 4. a. x2 1 4x 1 4 b. x2 2 6x 1 9 c. x2 2 16 5. 2x2 2 x"3 2 3

6. 2x3 1 7x2 1 5x 2 2 7. x 1 4"x 1 4

x 2 4 8.

1

"4 1 h 1 2

9. 3x 2 1 10. 3x2 2 2x 1x 1 1x2 1 1

Exercises A.4Getting readyYou should be able to complete these vocabulary and concept statements before you proceed to the practice exercises.

Fill in the blanks. 1. A is a real number or the product of a

real number and one or more .

2. The of a monomial is the sum of the expo-nents of its .

3. A is a polynomial with three terms.

4. A is a polynomial with two terms.

5. A monomial is a polynomial with term.

6. The constant 0 is called the polynomial.

7. Terms with the same variables with the same expo-nents are called terms.

8. The of a polynomial is the same as the degree of its term of highest degree.

9. To combine like terms, we add their and keep the same and the same expo-nents.

10. The conjugate of 3"x 1 2 is .

Determine whether the given expression is a polynomial. If so, tell whether it is a monomial, a binomial, or a trino-mial, and give its degree.

11. x2 1 3x 1 4

12. 5xy 2 x3

13. x3 1 y1/2

14. x23 2 5y22

15. 4x2 2 "5x3

16. x2y3

17. "15

18. 5x

1x5

1 5

19. 0

20. 3y3 2 4y2 1 2y 1 2

PracticePerform the operations and simplify.

21. 1x3 2 3x22 1 15x3 2 8x2 22. 12x4 2 5x32 1 17x3 2 x4 1 2x2 23. 1y5 1 2y3 1 72 2 1y5 2 2y3 2 72 24. 13t7 2 7t3 1 32 2 17t7 2 3t3 1 72

25. 2 1x2 1 3x 2 12 2 3 1x2 1 2x 2 42 1 4

26. 5 1x3 2 8x 1 32 1 2 13x2 1 5x2 2 7

27. 8 1t2 2 2t 1 52 1 4 1t2 2 3t 1 22 2 6 12t2 2 82

28. 23 1x3 2 x2 1 2 1x2 1 x2 1 3 1x3 2 2x2 29. y 1y2 2 12 2 y2 1y 1 22 2 y 12y 2 22 30. 24a2 1a 1 12 1 3a 1a2 2 42 2 a2 1a 1 22

31. xy 1x 2 4y2 2 y 1x2 1 3xy2 1 xy 12x 1 3y2

32. 3mn 1m 1 2n2 2 6m 13mn 1 12 2 2n 14mn 2 12

33. 2x2y3 14xy42 34. 215a3b 122a2b32

35. 23m2n 12mn22 a2mn12

b

36. 23r2s3

5a2r2s

3b a15rs2

2b

37. 24rs 1r2 1 s22 38. 6u2v 12uv2 2 y2 39. 6ab2c 12ac 1 3bc2 2 4ab2c2

40. 2mn2

214mn 2 6m2 2 82

41. 1a 1 22 1a 1 22 42. 1y 2 52 1y 2 52

43. 1a 2 62 2 44. 1t 1 92 2

45. 1x 1 42 1x 2 42 46. 1z 1 72 1z 2 72

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SectionA.4PolynomialsA51

47. 1x 2 32 1x 1 52 48. 1z 1 42 1z 2 62

49. 1u 1 22 13u 2 22 50. 14x 1 12 12x 2 32

51. 15x 2 12 12x 1 32 52. 14x 2 12 12x 2 72

53. 13a 2 2b2 2 54. 14a 1 5b2 14a 2 5b2

55. 13m 1 4n2 13m 2 4n2 56. 14r 1 3s2 2

57. 12y 2 4x2 13y 2 2x2 58. 122x 1 3y2 13x 1 y2

59. 19x 2 y2 1x2 2 3y2 60. 18a2 1 b2 1a 1 2b2

61. 15z 1 2t2 1z2 2 t2 62. 1y 2 2x22 1x2 1 3y2

63. Q"5 1 3xRQ2 2 "5xR 64. Q"2 1 xRQ3 1 "2xR 65. 13x 2 12 3 66. 12x 2 32 3

67. 13x 1 12 12x2 1 4x 2 32 68. 12x 2 52 1x2 2 3x 1 22 69. 13x 1 2y2 12x2 2 3xy 1 4y22 70. 14r 2 3s2 12r2 1 4rs 2 2s22

Multiply the expressions as you would multiply polynomials.

71. 2yn 13yn 1 y2n2 72. 3a2n 12an 1 3an212 73. 25x2nyn 12x2ny2n 1 3x22nyn2 74. 22a3nb2n 15a23nb 2 ab22n2 75. 1xn 1 32 1xn 2 42 76. 1an 2 52 1an 2 32 77. 12rn 2 72 13rn 2 22 78. 14zn 1 32 13zn 1 12 79. x1/2 1x1/2y 1 xy1/22 80. ab1/2 1a1/2b1/2 1 b1/22 81. 1a1/2 1 b1/22 1a1/2 2 b1/22 82. 1x3/2 1 y1/22 2

Rationalize each denominator.

83. 2

"3 2 1 84.

1

"5 1 2

85.3x

"7 1 2 86.

14y

"2 2 3

87.x

x 2 "3 88.

y

2y 1 "7

89.y 1 "2

y 2 "2 90.

x 2 "3

x 1 "3

91."2 2 "3

1 2 "3 92.

"3 2 "2

1 1 "2

93."x 2 "y

"x 1 "y 94.

"2x 1 y

"2x 2 y

Rationalize each numerator.

95. "2 1 1

2 96.

"x 2 33

97.y 2 "3

y 1 "3 98.

"a 2 "b

"a 1 "b

99. "x 1 3 2 "x

3 100.

"2 1 h 2 "2h

Perform each division and write all answers without using negative exponents.

101. 36a2b3

18ab6 102. 245r2s5t3

27r6s2t8

103. 16x6y4z9

224x9y6z0 104. 32m6n4p2

26m6n7p2

105.5x3y2 1 15x3y4

10x2y3 106. 9m4n9 2 6m3n4

12m3n3

107.24x5y7 2 36x2y5 1 12xy

60x5y4

108.9a3b4 1 27a2b4 2 18a2b3

18a2b7

Perform each division. If there is a nonzero remainder, write the answer in quotient 1 remainder

divisor form. 109. x 1 3q3x2 1 11x 1 6

110. 3x 1 2q3x2 1 11x 1 6

111. 2x 2 5q2x2 2 19x 1 37

112. x 2 7q2x2 2 19x 1 35

113. 2x3 1 1x 2 1

114. 2x3 2 9x2 1 13x 2 20

2x 2 7

115. x2 1 x 2 1qx3 2 2x2 2 4x 1 3

116. x2 2 3qx3 2 2x2 2 4x 1 5

117.x5 2 2x3 2 3x2 1 9

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A52 AppendixAAReviewofBasicAlgebra

118.x5 2 2x3 2 3x2 1 9

x3 2 3

119.x5 2 32x 2 2

120.x4 2 1x 1 1

121. 11x 2 10 1 6x2q36x4 2 121x2 1 120 1 72x3 2 142x

122. x 1 6x2 2 12q2121x2 1 72x3 2 142x 1 120 1 36x4

applications 123. Geometry Find an expression that represents the area

of the brick wall.

(x – 2) ft

(x + 5) ft

124. Geometry The area of the triangle shown in the illustration is represented as 1x2 1 3x 2 402 square feet. Find an expression that represents its height.

(x + 8) ft

Height

125. GiftBoxes The corners of a 12 in.-by-12 in. piece of cardboard are folded inward and glued to make a box. Write a polynomial that represents the volume of the resulting box.

x

x

x

x x

x

x

x

Crease here andfold inward

12 in.

12 in.

126. Travel Complete the following table, which shows the rate (mph), time traveled (hr), and distance traveled (mi) by a family on vacation.

r ? t 5 d

3x 1 4 3x2 1 19x 1 20

discovery and Writing 127. Show that a trinomial can be squared by using the

formula 1a 1 b 1 c2 2 5 a2 1 b2 1 c2 1 2ab 1 2bc 1 2ac.

128. Show that 1a 1 b 1 c 1 d2 2 5 a2 1 b2 1 c2 1

d 2 1 2ab 1 2ac 1 2ad 1 2bc 1 2bd 1 2cd .

129. Explain the FOIL method.

130. Explain how to rationalize the numerator of "x 1 2x .

131. Explain why 1a 1 b2 2 2 a2 1 b2.

132. Explain why "a2 1 b2 2 "a2 1 "b2.

reviewSimplify each expression. Assume that all variables repre-sent positive numbers.

133. 93/2 134. a 8125

b22/3

135. a625x4

16y8 b3/4

136. "80x4

137. "3 16ab4 2 b"3 54ab 138. x"4 1,280x 1 "4 80x5

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SectionA.5FactoringPolynomialsA53

A.5 Factoring PolynomialsInthissection,wewilllearnto

1.Factoroutacommonmonomial.2.Factorbygrouping.3.Factorthedifferenceoftwosquares.4.Factortrinomials.5.Factortrinomialsbygrouping.6.Factorthesumanddifferenceoftwocubes.7.Factormiscellaneouspolynomials.

The television network series CSI: Crime Scene Investigation and its spinoffs, CSI: NY and CSI: Miami, are favorite television shows of many people. Their fans enjoy watching a team of forensic scientists uncover the circumstances that led to an unusual death or crime. These mysteries are captivating because many viewers are interested in the field of forensic science.

In this section, we will investigate a mathematics mystery. We will be given a polynomial and asked to unveil or uncover the two or more polynomials that were multiplied together to obtain the given polynomial. The process that we will use to solve this mystery is called factoring.

When two or more polynomials are multiplied together, each one is called a factor of the resulting product. For example, the factors of 7 1x 1 22 1x 1 32 are

7, x 1 2, and x 1 3

The process of writing a polynomial as the product of several factors is called factoring.

In this section, we will discuss factoring where the coefficients of the polyno-mial and the polynomial factors are integers. If a polynomial cannot be factored by using integers only, we call it a primepolynomial.

1. Factor Out a Common MonomialThe simplest type of factoring occurs when we see a common monomial factor in each term of the polynomial. In this case, our strategy is to use the Distributive Property and factor out the greatest common factor.

eXaMPLe 1 Factoring by removing a Common Monomial

Factor: 3xy2 1 6x.

sOLUtiOn We note that each term contains a greatest common factor of 3x:

3xy2 1 6x 5 3x 1y22 1 3x 122We can then use the Distributive Property to factor out the common factor of 3x:

3xy2 1 6x 5 3x 1y2 1 22 We can check by multiplying: 3x 1y2 1 22 5 3xy2 1 6x.

self Check 1 Factor: 4a2 2 8ab.

now try exercise 11.

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A54 AppendixAAReviewofBasicAlgebra

eXaMPLe 2 Factoring by removing a Common Monomial

Factor: x2y2z2 2 xyz.

sOLUtiOn We factor out the greatest common factor of xyz:

x2y2z2 2 xyz 5 xyz 1xyz2 2 xyz 112 5 xyz 1xyz 2 12 We can check by multiplying.

The last term in the expression x2y2z2 2 xyz has an understood coefficient of 1. When the xyz is factored out, the 1 must be written.

self Check 2 Factor: a2b2c2 1 a3b3c3.

now try exercise 13.

2. Factor by GroupingA strategy that is often helpful when factoring a polynomial with four or more terms is called grouping. Terms with common factors are grouped together and then their greatest common factors are factored out using the Distributive Property.

eXaMPLe 3 Factoring by Grouping

Factor: ax 1 bx 1 a 1 b.

sOLUtiOn Although there is no factor common to all four terms, we can factor x out of the first two terms and write the expression as

ax 1 bx 1 a 1 b 5 x 1a 1 b2 1 1a 1 b2We can now factor out the common factor of a 1 b.

ax 1 bx 1 a 1 b 5 x 1a 1 b2 1 1a 1 b2 5 x 1a 1 b2 1 1 1a 1 b2 5 1a 1 b2 1x 1 12 We can check by multiplying.

self Check 3 Factor: x2 1 xy 1 2x 1 2y.

now try exercise 17.

3. Factor the difference of two squaresA binomial that is the difference of the squares of two quantities factors easily. The strategy used is to write the polynomial as the product of two factors. The first factor is the sum of the quantities and the other factor is the difference of the quantities.

eXaMPLe 4 Factoring the difference of two squares

Factor: 49x2 2 4.

sOLUtiOn We observe that each term is a perfect square:

49x2 2 4 5 17x2 2 2 22

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SectionA.5FactoringPolynomialsA55

Factoring the difference of two squares

The difference of the squares of two quantities is the product of two factors. One is the sum of the quantities, and the other is the difference of the quantities. Thus, 49x2 2 4 factors as

49x2 2 4 5 17x2 2 2 22

5 17x 1 22 17x 2 22 We can check by multiplying.

self Check 4 Factor: 9a2 2 16b2.

now try exercise 21.

Example 4 suggests a formula for factoring the difference of two squares.

x2 2 y2 5 1x 1 y2 1x 2 y2

eXaMPLe 5 Factoring the difference of two squares twice in One Problem

Factor: 16m4 2 n4.

sOLUtiOn The binomial 16m4 2 n4 can be factored as the difference of two squares:

16m4 2 n4 5 14m22 2 2 1n22 2

5 14m2 1 n22 14m2 2 n22The first factor is the sum of two squares and is prime. The second factor is a dif-ference of two squares and can be factored:

16m4 2 n4 5 14m2 1 n22 3 12m2 2 2 n2 4 5 14m2 1 n22 12m 1 n2 12m 2 n2 We can check by multiplying.

self Check 5 Factor: a4 2 81b4.

now try exercise 23.

eXaMPLe 6 removing a Common Factor and Factoring the difference of two squares

Factor: 18t2 2 32.

sOLUtiOn We begin by factoring out the common monomial factor of 2.

18t2 2 32 5 2 19t2 2 162 Since 9t2 2 16 is the difference of two squares, it can be factored.

18t2 2 32 5 2 19t2 2 162 5 2 13t 1 42 13t 2 42 We can check by multiplying.

self Check 6 Factor: 23x2 1 12.

now try exercise 61.

Caution

Ifyouarelimitedtointegercoefficients,thesumoftwosquarescannotbefactored.Forexample,x2 1 y2isaprimepolynomial.

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A56 AppendixAAReviewofBasicAlgebra

Factoring trinomial squares

4. Factor trinomialsTrinomials that are squares of binomials can be factored by using the following formulas.

(1) x2 1 2xy 1 y2 5 1x 1 y2 1x 1 y2 5 1x 1 y2 2

(2) x2 2 2xy 1 y2 5 1x 2 y2 1x 2 y2 5 1x 2 y2 2

For example, to factor a2 2 6a 1 9, we note that it can be written in the form

a2 2 2 13a2 1 32 x 5 a and y 5 3

which matches the left side of Equation 2 above. Thus,

a2 2 6a 1 9 5 a2 2 2 13a2 1 32

5 1a 2 32 1a 2 32 5 1a 2 32 2 We can check by multiplying.

Factoring trinomials that are not squares of binomials often requires some guesswork. If a trinomial with no common factors is factorable, it will factor into the product of two binomials.

eXaMPLe 7 Factoring a trinomial with Leading Coefficient of 1

Factor: x2 1 3x 2 10.

sOLUtiOn To factor x2 1 3x 2 10, we must find two binomials x 1 a and x 1 b such that

x2 1 3x 2 10 5 1x 1 a2 1x 1 b2where the product of a and b is –10 and the sum of a and b is 3.

ab 5 210 and a 1 b 5 3

To find such numbers, we list the possible factorizations of –10:

10 1212 5 1222 210 112 25 122Only in the factorization 5 1222 do the factors have a sum of 3. Thus, a 5 5 and b 5 22, and

x2 1 3x 2 10 5 1x 1 a2 1x 1 b2 (3) x2 1 3x 2 10 5 1x 1 52 1x 2 22 We can check by multiplying.

Because of the Commutative Property of Multiplication, the order of the fac-tors in Equation 3 is not important. Equation 3 can also be written as

x2 1 3x 2 10 5 1x 2 22 1x 1 52

self Check 7 Factor: p2 2 5p 2 6.

now try exercise 29.

eXaMPLe 8 Factoring a trinomial with Leading Coefficient not 1

Factor: 2x2 2 x 2 6.

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SectionA.5FactoringPolynomialsA57

strategy for Factoring a General trinomial

with integer Coefficients

sOLUtiOn Since the first term is 2x2, the first terms of the binomial factors must be 2x and x:

2x2 2 x 2 6 5 12x 2 1x 2The product of the last terms must be 26, and the sum of the products of the outer terms and the inner terms must be 2x. Since the only factorization of 26 that will cause this to happen is 3 1222 , we have

2x2 2 x 2 6 5 12x 1 32 1x 2 22 We can check by multiplying.

self Check 8 Factor: 6x2 2 x 2 2.

now try exercise 33.

It is not easy to give specific rules for factoring trinomials, because some guess-work is often necessary. However, the following hints are helpful.

1. Write the trinomial in descending powers of one variable.

2. Factor out any greatest common factor, including 21 if that is necessary to make the coefficient of the first term positive.

3. When the sign of the first term of a trinomial is 1 and the sign of the third term is 1, the sign between the terms of each binomial factor is the same as the sign of the middle term of the trinomial.

When the sign of the first term is 1 and the sign of the third term is 2, one of the signs between the terms of the binomial factors is 1 and the other is 2.

4. Try various combinations of first terms and last terms until you find one that works. If no possibilities work, the trinomial is prime.

5. Check the factorization by multiplication.

eXaMPLe 9 Factoring a trinomial Completely

Factor: 10xy 1 24y2 2 6x2.

sOLUtiOn We write the trinomial in descending powers of x and then factor out the common factor of 22.

10xy 1 24y2 2 6x2 5 26x2 1 10xy 1 24y2

5 22 13x2 2 5xy 2 12y22Since the sign of the third term of 3x2 2 5xy 2 12y2 is 2, the signs between

the binomial factors will be opposite. Since the first term is 3x2, the first terms of the binomial factors must be 3x and x:

22 13x2 2 5xy 2 12y22 5 22 13x 2 1x 2The product of the last terms must be 212y2, and the sum of the outer terms

and the inner terms must be 25xy. Of the many factorizations of 212y2, only 4y 123y2 leads to a middle term of 25xy. So we have

10xy 1 24y2 2 6x2 5 26x2 1 10xy 1 24y2

5 22 13x2 2 5xy 2 12y22 5 22 13x 1 4y2 1x 2 3y2 We can check by multiplying.

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A58 AppendixAAReviewofBasicAlgebra

self Check 9 Factor: 26x2 2 15xy 2 6y2.

now try exercise 69.

5. Factor trinomials by GroupingAnother way of factoring trinomials involves factoring by grouping. This method can be used to factor trinomials of the form ax2 1 bx 1 c. For example, to factor 6x2 1 5x 2 6, we note that a 5 6, b 5 5, and c 5 26, and proceed as follows:

1. Find the product ac: 6 1262 5 236. This number is called the keynumber. 2. Find two factors of the key number 12362 whose sum is b 5 5. Two such num-

bers are 9 and 24.

9 1242 5 236 and 9 1 1242 5 5

3. Use the factors 9 and 24 as coefficients of two terms to be placed between 6x2 and 26.

6x2 1 5x 2 6 5 6x2 1 9x 2 4x 2 6

4. Factor by grouping:

6x2 1 9x 2 4x 2 6 5 3x 12x 1 32 2 2 12x 1 32 5 12x 1 32 13x 2 22 Factor out 2x 1 3.

eXaMPLe 10 Factoring a trinomial by Grouping

Factor: 15x2 1 x 2 2.

sOLUtiOn Since a 5 15 and c 5 22 in the trinomial, ac 5 230. We now find factors of 230 whose sum is b 5 1. Such factors are 6 and 25. We use these factors as coefficients of two terms to be placed between 15x2 and –2.

15x2 1 6x 2 5x 2 2

Finally, we factor by grouping.

3x 15x 1 22 2 15x 1 22 5 15x 1 22 13x 2 12

self Check 10 Factor: 15a2 1 17a 2 4.

now try exercise 39.

We can often factor polynomials with variable exponents. For example, if n is a natural number,

a2n 2 5an 2 6 5 1an 1 12 1an 2 62because

1an 1 12 1an 2 62 5 a2n 2 6an 1 an 2 6

5 a2n 2 5an 2 6 Combine like terms.

6. Factor the sum and difference of two CubesTwo other types of factoring involve binomials that are the sum or the difference of two cubes. Like the difference of two squares, they can be factored by using a formula.

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SectionA.5FactoringPolynomialsA59

Factoring the sum and difference of two Cubes

x3 1 y3 5 1x 1 y2 1x2 2 xy 1 y22x3 2 y3 5 1x 2 y2 1x2 1 xy 1 y22

eXaMPLe 11 Factoring the sum of two Cubes

Factor: 27x6 1 64y3.

sOLUtiOn We can write this expression as the sum of two cubes and factor it as follows:

27x6 1 64y3 5 13x22 3 1 14y2 3

5 13x2 1 4y2 3 13x22 2 2 13x22 14y2 1 14y2 2 4 5 13x2 1 4y2 19x4 2 12x2y 1 16y22 We can check by multiplying.

self Check 11 Factor: 8a3 1 1,000b6.

now try exercise 41.

eXaMPLe 12 Factoring the difference of two Cubes

Factor: x3 2 8.

sOLUtiOn This binomial can be written as x3 2 23, which is the difference of two cubes. Sub-stituting into the formula for the difference of two cubes gives

x3 2 23 5 1x 2 22 1x2 1 2x 1 222 5 1x 2 22 1x2 1 2x 1 42 We can check by multiplying.

self Check 12 Factor: p3 2 64.

now try exercise 43.

7. Factor Miscellaneous Polynomials

eXaMPLe 13 Factoring a Miscellaneous Polynomial

Factor: x2 2 y2 1 6x 1 9.

sOLUtiOn Here we will factor a trinomial and a difference of two squares.

x2 2 y2 1 6x 1 9 5 x2 1 6x 1 9 2 y2 Use the Commutative Property to rearrange the terms.

5 1x 1 32 2 2 y2 Factor x2 1 6x 1 9.

5 1x 1 3 1 y2 1x 1 3 2 y2 Factor the difference of two squares.

We could try to factor this expression in another way.

x2 2 y2 1 6x 1 9 5 1x 1 y2 1x 2 y2 1 3 12x 1 32 Factor x2 2 y2 and 6x 1 9.

However, we are unable to finish the factorization. If grouping in one way doesn’t work, try various other ways.

self Check 13 Factor: a2 1 8a 2 b2 1 16.

now try exercise 97.Not For Sale

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A60 AppendixAAReviewofBasicAlgebra

Factoring strategy

eXaMPLe 14 Factoring a Miscellaneous trinomial

Factor: z4 2 3z2 1 1.

sOLUtiOn This trinomial cannot be factored as the product of two binomials, because no combination will give a middle term of 23z2. However, if the middle term were 22z2, the trinomial would be a perfect square, and the factorization would be easy:

z4 2 2z2 1 1 5 1z2 2 12 1z2 2 12 5 1z2 2 12 2

We can change the middle term in z4 2 3z2 1 1 to 22z2 by adding z2 to it. However, to make sure that adding z2 does not change the value of the trinomial, we must also subtract z2. We can then proceed as follows.

z4 2 3z2 1 1 5 z4 2 3z2 1 z2 1 1 2 z2 Add and subtract z2.

5 z4 2 2z2 1 1 2 z2 Combine 23z2 and z2.

5 1z2 2 12 2 2 z2 Factor z4 2 2z2 1 1.

5 1z2 2 1 1 z2 1z2 2 1 2 z2 Factor the difference of two squares.

In this type of problem, we will always try to add and subtract a perfect square in hopes of making a perfect-square trinomial that will lead to factoring a differ-ence of two squares.

self Check 14 Factor: x4 1 3x2 1 4.

now try exercise 103.

It is helpful to identify the problem type when we must factor polynomials that are given in random order.

1. Factor out all common monomial factors.

2. If an expression has two terms, check whether the problem type is

a. The difference of two squares:

x2 2 y2 5 1x 1 y2 1x 2 y2 b. The sum of two cubes:

x3 1 y3 5 1x 1 y2 1x2 2 xy 1 y22 c. The difference of two cubes:

x3 2 y3 5 1x 2 y2 1x2 1 xy 1 y22 3. If an expression has three terms, try to factor it as a trinomial.

4. If an expression has four or more terms, try factoring by grouping.

5. Continue until each individual factor is prime.

6. Check the results by multiplying.

self Check answers 1. 4a 1a 2 2b2 2. a2b2c2 11 1 abc2 3. 1x 1 y2 1x 1 22 4. 13a 1 4b2 13a 2 4b2 5. 1a2 1 9b22 1a 1 3b2 1a 2 3b2 6. 23 1x 1 22 1x 2 22 7. 1p 2 62 1p 1 12 8. 13x 2 22 12x 1 12 9. 23 1x 1 2y2 12x 1 y2 10. 13a 1 42 15a 2 12 11. 8 1a 1 5b22 1a2 2 5ab2 1 25b42 12. 1p 2 42 1p2 1 4p 1 162 13. 1a 1 4 1 b2 1a 1 4 2 b2 14. 1x2 1 2 1 x2 1x2 1 2 2 x2

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SectionA.5FactoringPolynomialsA61

Exercises A.5Getting readyYou should be able to complete these vocabulary and concept statements before you proceed to the practice exercises.

Fill in the blanks. 1. When polynomials are multiplied together, each

polynomial is a of the product.

2. If a polynomial cannot be factored using coefficients, it is called a polynomial.

Complete each factoring formula.

3. ax 1 bx 5

4. x2 2 y2 5

5. x2 1 2xy 1 y2 5

6. x2 2 2xy 1 y2 5

7. x3 1 y3 5

8. x3 2 y3 5

Practice In each expression, factor out the greatest common monomial.

9. 3x 2 6 10. 5y 2 15

11. 8x2 1 4x3 12. 9y3 1 6y2

13. 7x2y2 1 14x3y2 14. 25y2z 2 15yz2

In each expression, factor by grouping.

15. a 1x 1 y2 1 b 1x 1 y2 16. b 1x 2 y2 1 a 1x 2 y2 17. 4a 1 b 2 12a2 2 3ab 18. x2 1 4x 1 xy 1 4y

In each expression, factor the difference of two squares.19. 4x2 2 9 20. 36z2 2 49 21. 4 2 9r2 22. 16 2 49x2 23. 81x4 2 1 24. 81 2 x4 25. 1x 1 z2 2 2 25 26. 1x 2 y2 2 2 9

In each expression, factor the trinomial. 27. x2 1 8x 1 16 28. a2 2 12a 1 36 29. b2 2 10b 1 25 30. y2 1 14y 1 49 31. m2 1 4mn 1 4n2 32. r2 2 8rs 1 16s2

33. 12x2 2 xy 2 6y2 34. 8x2 2 10xy 2 3y2

In each expression, factor the trinomial by grouping. 35. x2 1 10x 1 21 36. x2 1 7x 1 10 37. x2 2 4x 2 12 38. x2 2 2x 2 63 39. 6p2 1 7p 2 3 40. 4q2 2 19q 1 12

In each expression, factor the sum of two cubes. 41. t3 1 343 42. r3 1 8s3

In each expression, factor the difference of two cubes. 43. 8z3 2 27 44. 125a3 2 64

Factor each expression completely. If an expression is prime, so indicate.

45. 3a2bc 1 6ab2c 1 9abc2

46. 5x3y3z3 1 25x2y2z2 2 125xyz

47. 3x3 1 3x2 2 x 2 1

48. 4x 1 6xy 2 9y 2 6

49. 2txy 1 2ctx 2 3ty 2 3ct

50. 2ax 1 4ay 2 bx 2 2by

51. ax 1 bx 1 ay 1 by 1 az 1 bz

52. 6x2y3 1 18xy 1 3x2y2 1 9x

53. x2 2 1y 2 z2 2 54. z2 2 1y 1 32 2 55. 1x 2 y2 2 2 1x 1 y2 2 56. 12a 1 32 2 2 12a 2 32 2 57. x4 2 y4 58. z4 2 81 59. 3x2 2 12 60. 3x3y 2 3xy 61. 18xy2 2 8x 62. 27x2 2 12 63. x2 2 2x 1 15 64. x2 1 x 1 2

65. 215 1 2a 1 24a2 66. 232 2 68x 1 9x2 67. 6x2 1 29xy 1 35y2 68. 10x2 2 17xy 1 6y2

69. 12p2 2 58pq 2 70q2 70. 3x2 2 6xy 2 9y2

71. 26m2 1 47mn 2 35n2 72. 214r2 2 11rs 1 15s2

73. 26x3 1 23x2 1 35x 74. 2y3 2 y2 1 90y

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A62 AppendixAAReviewofBasicAlgebra

75. 6x4 2 11x3 2 35x2 76. 12x 1 17x2 2 7x3

77. x4 1 2x2 2 15 78. x4 2 x2 2 6

79. a2n 2 2an 2 3 80. a2n 1 6an 1 8

81. 6x2n 2 7xn 1 2 82. 9x2n 1 9xn 1 2

83. 4x2n 2 9y2n 84. 8x2n 2 2xn 2 3

85. 10y2n 2 11yn 2 6 86. 16y4n 2 25y2n

87. 2x3 1 2,000 88. 3y3 1 648

89. 1x 1 y2 3 2 64

90. 1x 2 y2 3 1 27

91. 64a6 2 y6

92. a6 1 b6

93. a3 2 b3 1 a 2 b

94. 1a2 2 y22 2 5 1a 1 y2 95. 64x6 1 y6

96. z2 1 6z 1 9 2 225y2

97. x2 2 6x 1 9 2 144y2

98. x2 1 2x 2 9y2 1 1

99. 1a 1 b2 2 2 3 1a 1 b2 2 10

100. 2 1a 1 b2 2 2 5 1a 1 b2 2 3

101. x6 1 7x3 2 8

102. x6 2 13x4 1 36x2

103. x4 1 x2 1 1

104. x4 1 3x2 1 4

105. x4 1 7x2 1 16

106. y4 1 2y2 1 9

107. 4a4 1 1 1 3a2

108. x4 1 25 1 6x2

109. Candy To find the amount of chocolate used in the outer coating of one of the malted-milk balls shown, we can find the volume V of the chocolate shell using the formula V 5 4

3pr13 2 4

3pr23. Factor

the expression on the right side of the formula.

Outer radius r1

Inner radius r2

110. MovieStunts The formula that gives the distance a stuntwoman is above the ground t seconds after she falls over the side of a 144-foot tall building is f 5 144 2 16t2. Factor the right side.

144 ft

discovery and Writing 111. Explain how to factor the difference of two squares.

112. Explain how to factor the difference of two cubes.

113. Explain how to factor a2 2 b2 1 a 1 b.

114. Explain how to factor x2 1 2x 1 1.

Factor the indicated monomial from the given expression.

115. 3x 1 2; 2 116. 5x 2 3; 5

117. x2 1 2x 1 4; 2 118. 3x2 2 2x 2 5; 3

119. a 1 b;a 120. a 2 b;b

121. x 1 x1/2; x1/2 122. x3/2 2 x1/2; x1/2

123. 2x 1 "2y; "2 124. "3a 2 3b; "3

125. ab3/2 2 a3/2b; ab 126. ab2 1 b; b21

Factor each expression by grouping three terms and two terms.

127. x2 1 x 2 6 1 xy 2 2y

128. 2x2 1 5x 1 2 2 xy 2 2y

129. a4 1 2a3 1 a2 1 a 1 1

130. a4 1 a3 2 2a2 1 a 2 1

review 131. Which natural number is neither prime nor

composite?

132. Graph the interval 322,32 . 133. Simplify: 1x3x22 4.

134. Simplify: 1a32 3 1a22 4

1a2a32 3 . 135. a 3x4x3

6x22x4b0

136. Simplify: "20x5.

137. Simplify: "20x 2 "125x.

138. Rationalize the denominator: 3

!3 3.

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SectionA.6RationalExpressionsA63

A.6 Rational ExpressionsInthissection,wewilllearnto

1.Definerationalexpressions.2.Simplifyrationalexpressions.3.Multipleanddividerationalexpressions.4.Addandsubtractrationalexpressions.5.Simplifycomplexfractions.

Abercrombie and Fitch (A&F) is a very successful American clothing company founded in 1892 by David Abercrombie and Ezra Fitch. Today the stores are popu-lar shopping destinations for university students wanting to keep up with the latest styles and trends.

Suppose that a clothing manufacturer finds that the cost in dollars of produc-ing x fleece vintage shirts is given by the algebraic expression 13x 1 1,000. The average cost of producing each shirt could be obtained by dividing the production cost, 13x 1 1,000, by the number of shirts produced, x. The algebraic fraction

13x 1 1,000x

represents the average cost per shirt. We see that the average cost of producing 200 shirts would be $18.

13 12002 1 1,000200

5 18

An understanding of algebraic fractions is important in solving many real-life problems.

1. define rational expressionsIf x and y are real numbers, the quotient xy 1y 2 02 is called a fraction. The number x is called the numerator, and the number y is called the denominator.

Algebraic fractions are quotients of algebraic expressions. If the expressions are polynomials, the fraction is called a rationalexpression. The first two of the following algebraic fractions are rational expressions. The third is not, because the numerator and denominator are not polynomials.

5y2 1 2yy2 2 3y 2 7

8ab2 2 16c3

2x 1 3

x1/2 1 4x

x3/2 2 x1/2

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es

CautionRememberthatthedenominatorofafractioncannotbezero.

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A64 AppendixAAReviewofBasicAlgebra

Properties of Fractions

We summarize some of the properties of fractions as follows:

If a, b, c, and d are real numbers and no denominators are 0, then

EqualityofFractionsab

5cd

if and only if ad 5 bc

FundamentalPropertyofFractions

axbx

5ab

MultiplicationandDivisionofFractions

ab

?cd

5acbd

and ab

4cd

5ab

?dc

5adbc

AdditionandSubtractionofFractions

ab

1cb

5a 1 c

b and

ab

2cb

5a 2 c

b

The first two examples illustrate each of the previous properties of fractions.

eXaMPLe 1 illustrating the Properties of Fractions

Assume that no denominators are 0.

a. 2a3

54a6

Because 2a 162 5 3 14a2

b. 6xy10xy

53 12xy25 12xy2 Factor the numerator and denominator and divide out the common factors.

535

self Check 1 a. Is 3y5

515z25

? b. Simplify: 15a2b25ab2.

now try exercise 9.

eXaMPLe 2 illustrating the Properties of Fractions

Assume that no denominators are 0.

a. 2r7s

?3r5s

52r ? 3r7s ? 5s

b. 3mn4pq

42pq7mn

53mn4pq

?7mn2pq

56r2

35s2 521m2n2

8p2q2

c. 2ab5xy

1ab

5xy5

2ab 1 ab5xy

d. 6uv2

7w2 23uv2

7w2 56uv2 2 3uv2

7w2

53ab5xy

53uv2

7w2

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SectionA.6RationalExpressionsA65

self Check 2 Perform each operation:

a. 3a5b

?2a7b

b. 2ab3rs

42rs4ab

c. 5pq3t

13pq3t

d. 5mn2

3w2

mn2

3w

now try exercise 19.

To add or subtract rational expressions with unlike denominators, we write each expression as an equivalent expression with a common denominator. We can then add or subtract the expressions. For example,

3x5

12x7

53x 1725 172 1

2x 1527 152

4a2

152

3a2

105

4a2 12215 122 2

3a2 13210 132

521x35

110x35

58a2

302

9a2

30

521x 1 10x

35 5

8a2 2 9a2

30

531x35

52a2

30

5 2a2

30

A rational expression is inlowestterms if all factors common to the numerator and the denominator have been removed. To simplifyarationalexpression means to write it in lowest terms.

2. simplify rational expressionsTo simplify rational expressions, we use the Fundamental Property of Fractions. This enables us to divide out all factors that are common to the numerator and the denominator.

eXaMPLe 3 simplifying a rational expression

Simplify: x2 2 9

x2 2 3x 1x 2 0, 32 .

sOLUtiOn We factor the difference of two squares in the numerator, factor out x in the denom-inator, and divide out the common factor of x 2 3.

x2 2 9

x2 2 3x5

1x 1 32 1x 2 32x 1x 2 32 x 2 3

x 2 35 1

5x 1 3

x

self Check 3 Simplify: a2 2 4a

a2 2 a 2 12 1a 2 4,232 .

now try exercise 23.

We will encounter the following properties of fractions in the next examples.Not For Sale

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A66 AppendixAAReviewofBasicAlgebra

Properties of Fractions If a and b represent real numbers and there are no divisions by 0, then

• a1

5 a • aa

5 1

• ab

52a2b

5 2a

2b5 2

2ab

• 2ab

5a

2b5

2ab

5 22a2b

eXaMPLe 4 simplifying a rational expression

Simplify: x2 2 2xy 1 y2

y 2 x 1x 2 y2 .

sOLUtiOn We factor the trinomial in the numerator, factor 21 from the denominator, and divide out the common factor of x 2 y.

x2 2 2xy 1 y2

y 2 x5

1x 2 y2 1x 2 y221 1x 2 y2

x 2 y

x 2 y5 1

5x 2 y

21

5 2x 2 y

1

5 2 1x 2 y2

self Check 4 Simplify: a2 2 ab 2 2b2

2b 2 a 12b 2 a 2 02 .

now try exercise 25.

eXaMPLe 5 simplifying a rational expression

Simplify: x2 2 3x 1 2x2 2 x 2 2

1x 2 2,212 .

sOLUtiOn We factor the numerator and denominator and divide out the common factor of x 2 2.

x2 2 3x 1 2x2 2 x 2 2

51x 2 12 1x 2 221x 1 12 1x 2 22 x 2 2

x 2 25 1

5x 2 1x 1 1

self Check 5 Simplify: a2 1 3a 2 4a2 1 2a 2 3

1a 2 1,232 .

now try exercise 27.

3. Multiply and divide rational expressions

eXaMPLe 6 Multiplying rational expressions

Multiply: x2 2 x 2 2

x2 2 1?

x2 1 2x 2 3x 2 2

1x 2 1, 21, 22 .

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SectionA.6RationalExpressionsA67

sOLUtiOn To multiply the rational expressions, we multiply the numerators, multiply the denominators, and divide out the common factors.

x2 2 x 2 2

x2 2 1?

x2 1 2x 2 3x 2 2

51x2 2 x 2 22 1x2 1 2x 2 32

1x2 2 12 1x 2 22

51x 2 22 1x 1 12 1x 2 12 1x 1 32

1x 1 12 1x 2 12 1x 2 22 x 2 2x 2 2

5 1, x 1 1x 1 1

5 1, x 2 1x 2 1

5 1

5 x 1 3

self Check 6 Simplify: x2 2 9x2 2 x

?x 2 1

x2 2 3x 1x 2 0, 1, 32 .

now try exercise 33.

eXaMPLe 7 dividing rational expressions

Divide: x2 2 2x 2 3

x2 2 44

x2 1 2x 2 15x2 1 3x 2 10

1x 2 2, 22, 3, 252 .

sOLUtiOn To divide the rational expressions, we multiply by the reciprocal of the second ratio-nal expression. We then simplify by factoring the numerator and denominator and dividing out the common factors.

x2 2 2x 2 3

x2 2 44

x2 1 2x 2 15x2 1 3x 2 10

5x2 2 2x 2 3

x2 2 4?

x2 1 3x 2 10x2 1 2x 2 15

51x2 2 2x 2 32 1x2 1 3x 2 102

1x2 2 42 1x2 1 2x 2 152

51x 2 32 1x 1 12 1x 2 22 1x 1 521x 1 22 1x 2 22 1x 1 52 1x 2 32 x 2 3

x 2 35 1,

x 2 2x 2 2

5 1, x 1 5x 1 5

5 1

5x 1 1x 1 2

self Check 7 Simplify: a2 2 aa 1 2

4a2 2 2aa2 2 4

1a 2 0, 2, 222 .

now try exercise 39.

eXaMPLe 8 Using Multiplication and division to simplify a rational expression

Simplify: 2x2 2 5x 2 3

3x 2 1?

3x2 1 2x 2 1x2 2 2x 2 3

42x2 1 x

3x ax 2

13

, 21, 3, 0, 212b.

sOLUtiOn We can change the division to a multiplication, factor, and simplify.

2x2 2 5x 2 33x 2 1

?3x2 1 2x 2 1x2 2 2x 2 3

42x2 1 x

3x

52x2 2 5x 2 3

3x 2 1?

3x2 1 2x 2 1x2 2 2x 2 3

?3x

2x2 1 x

512x2 2 5x 2 32 13x2 1 2x 2 12 13x213x 2 12 1x2 2 2x 2 32 12x2 1 x2

51x 2 32 12x 1 12 13x 2 12 1x 1 123x13x 2 12 1x 1 12 1x 2 32x 12x 1 12 x 2 3

x 2 35 1,

2x 1 12x 1 1

5 1, 3x 2 13x 2 1

5 1, x 1 1x 1 1

5 1, xx

5 1

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A68 AppendixAAReviewofBasicAlgebra

self Check 8 Simplify: x2 2 25x 2 2

4x2 2 5xx2 2 2x

?x2 1 2xx2 1 5x

1x 2 0, 2, 5, 252 .

now try exercise 45.

4. add and subtract rational expressionsTo add (or subtract) rational expressions with like denominators, we add (or sub-tract) the numerators and keep the common denominator.

eXaMPLe 9 adding rational expression with Like deonominators

Add: 2x 1 5x 1 5

13x 1 20x 1 5

1x 2 252 .

sOLUtiOn 2x 1 5x 1 5

13x 1 20x 1 5

55x 1 25x 1 5

Add the numerators and keep the common denominator.

55 1x 1 521 1x 1 52 Factor out 5 and divide out the common factor of x 1 5.

5 5

self Check 9 Add: 3x 2 2x 2 2

1x 2 6x 2 2

1x 2 22 .

now try exercise 49.

To add (or subtract) rational expressions with unlike denominators, we must find a common denominator, called the least (or lowest) commondenominator(LCD). Suppose the unlike denominators of three rational expressions are 12, 20, and 35. To find the LCD, we first find the prime factorization of each number.

12 5 4 ? 3 20 5 4 ? 5 35 5 5 ? 7

5 22 ? 3 5 22 ? 5

Because the LCD is the smallest number that can be divided by 12, 20, and 35, it must contain factors of 22, 3, 5, and 7. Thus, the

LCD 5 22 ? 3 ? 5 ? 7 5 420

That is, 420 is the smallest number that can be divided without remainder by 12, 20, and 35.

When finding an LCD, we always factor each denominator and then create the LCD by using each factor the greatest number of times that it appears in any one denominator. The product of these factors is the LCD.

This rule also applies if the unlike denominators of the rational expressions contain variables. Suppose the unlike denominators are x2 1x 2 52 and x 1x 2 52 3. To find the LCD, we use x2 and 1x 2 52 3. Thus, the LCD is the product x2 1x 2 52 3 .

eXaMPLe 10 adding rational expressions with Unlike denominators

Add: 1

x2 2 41

2x2 2 4x 1 4

1x 2 2, 222 .

Comment

Remembertofindtheleast commondenominator,useeachfactorthegreatestnumberoftimesthatitoccurs.

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SectionA.6RationalExpressionsA69

sOLUtiOn We factor each denominator and find the LCD.

x2 2 4 5 1x 1 22 1x 2 22x2 2 4x 1 4 5 1x 2 22 1x 2 22 5 1x 2 22 2

The LCD is 1x 1 22 1x 2 22 2. We then write each rational expression with its de-nominator in factored form, convert each rational expression into an equivalent ex-pression with a denominator of 1x 1 22 1x 2 22 2, add the expressions, and simplify.

1

x2 2 41

2x2 2 4x 1 4

51

1x 1 22 1x 2 22 12

1x 2 22 1x 2 22

51 1x 2 22

1x 1 22 1x 2 22 1x 2 22 12 1x 1 22

1x 2 22 1x 2 22 1x 1 22 x 2 2x 2 2

5 1, x 1 2x 1 2

5 1

51 1x 2 22 1 2 1x 1 221x 1 22 1x 2 22 1x 2 22

5x 2 2 1 2x 1 4

1x 1 22 1x 2 22 1x 2 22

53x 1 2

1x 1 22 1x 2 22 2

self Check 10 Add: 3

x2 2 6x 1 91

1x2 2 9

1x 2 3,232 .

now try exercise 59.

eXaMPLe 11 Combining and simplifying rational expressions with Unlike denominators

Simplify: x 2 2x2 2 1

2x 1 3

x2 1 3x 1 21

3x2 1 x 2 2

1x 2 1, 21, 222 .

sOLUtiOn We factor the denominators to find the LCD.

x2 2 1 5 1x 1 12 1x 2 12x2 1 3x 1 2 5 1x 1 22 1x 1 12x2 1 x 2 2 5 1x 1 22 1x 2 12

The LCD is 1x 1 12 1x 1 22 1x 2 12 . We now write each rational expression as an equivalent expression with this LCD, and proceed as follows:

x 2 2x2 2 1

2x 1 3

x2 1 3x 1 21

3x2 1 x 2 2

5x 2 2

1x 1 12 1x 2 12 2x 1 3

1x 1 12 1x 1 22 13

1x 2 12 1x 1 22 5

1x 2 22 1x 1 221x 1 12 1x 2 12 1x 1 22 2

1x 1 32 1x 2 121x 1 12 1x 1 22 1x 2 12 1

3 1x 1 121x 2 12 1x 1 22 1x 1 12 x 1 2

x 1 25 1,

x 2 1x 2 1

5 1, x 1 1x 1 1

5 1

51x2 2 42 2 1x2 1 2x 2 32 1 13x 1 32

1x 1 12 1x 1 22 1x 2 12

5x2 2 4 2 x2 2 2x 1 3 1 3x 1 3

1x 1 12 1x 1 22 1x 2 12

5x 1 2

1x 1 12 1x 1 22 1x 2 12 5

11x 1 12 1x 2 12 Divide out the common factor of x 1 2.

x 1 2x 1 2

5 1

Comment

Alwaysattempttosimplifythefinalresult.Inthiscase,thefinalfractionisalreadyinlowestterms.

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A70 AppendixAAReviewofBasicAlgebra

strategies for simplifying Complex Fractions

self Check 11 Simplify: 4y

y2 2 12

2y 1 1

1 2 1y 2 1, 212 .

now try exercise 61.

5. simplify Complex FractionsA complexfraction is a fraction that has a fraction in its numerator or a fraction in its denominator. There are two methods generally used to simplify complex frac-tions. These are stated here for you.

Method1:MultiplytheComplexFractionby1

• Determine the LCD of all fractions in the complex fraction. • Multiply both numerator and denominator of the complex fraction by the

LCD. Note that when we multiply by LCDLCD we are multiplying by 1.

Method2: SimplifytheNumeratorandDenominatorandthenDivide

• Simplify the numerator and denominator so that both are single fractions.

• Perform the division by multiplying the numerator by the reciprocal of the denominator.

eXaMPLe 12 simplifying Complex Fractions

Simplify:

1x

11y

xy

1x, y 2 02 .

Method1: We note that the LCD of the three fractions in the complex fraction is xy. So we multiply the numerator and denominator of the complex fraction by xy and simplify:

1x

11y

xy

5

xya1x

11yb

xyaxyb

5

xyx

1xyy

xyxy

5y 1 x

x2

Method2: We combine the fractions in the numerator of the complex fraction to obtain a single fraction over a single fraction.

1x

11y

xy

5

1 1y2x 1y2 1

1 1x2y 1x2

xy

5

y 1 xxyxy

Then we use the fact that any fraction indicates a division:

y 1 xxyxy

5y 1 x

xy4

xy

5y 1 x

xy?

yx

51y 1 x2y

xyx5

y 1 xx2

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SectionA.6RationalExpressionsA71

self Check 12 Simplify:

1x

21y

1x

11y

1x, y 2 02 .

now try exercise 81.

self Check answers 1. a. no b. 3a5b

2. a. 6a2

35b2 b. 4a2b2

3r2s2 c. 8pq3t

d. 4mn2

3w

3. a

a 1 3 4. 2 1a 1 b2 5.

a 1 4a 1 3

6. x 1 3

x2 7. a 2 1

8. x 1 2 9. 4 10. 4x 1 6

1x 1 32 1x 2 32 2 11. 2y

y 2 1 12.

y 2 xy 1 x

Exercises A.6Getting readyYou should be able to complete these vocabulary and concept statements before you proceed to the practice exercises.

Fill in the blanks. 1. In the fraction ab, a is called the .

2. In the fraction ab, b is called the .

3. ab 5 c

d if and only if .

4. The denominator of a fraction can never be .

Complete each formula.

5. ab

?cd

5 6. ab

4cd

5

7. ab

1cb

5 8. ab

2cb

5

Determine whether the fractions are equal. Assume that no denominators are 0.

9. 8x3y

, 16x6y

10. 3x2

4y2, 12y2

16x2

11. 25xyz12ab2c

, 50a2bc24xyz

12. 15rs2

4rs2 , 37.5a3

10a3

Practice Simplify each rational expression. Assume that no de-nominators are 0.

13. 7a2b21ab2 14.

35p3q2

49p4q

Perform the operations and simplify, whenever possible. Assume that no denominators are 0.

15. 4x7

?25a

16. 25y2z

?4y2

17. 8m5n

43m10n

18. 15p8q

425p16q2

19. 3z5c

12z5c

20. 7a4b

23a4b

21. 15x2y7a2b3 2

x2y7a2b3 22.

8rst2

15m4t2 17rst2

15m4t2

Simplify each fraction. Assume that no denominators are 0.

23. 2x 2 4x2 2 4

24. x2 2 16

x2 2 8x 1 16

25. 4 2 x2

x2 2 5x 1 6 26.

25 2 x2

x2 1 10x 1 25

27. 6x3 1 x2 2 12x4x3 1 4x2 2 3x

28. 6x4 2 5x3 2 6x2

2x3 2 7x2 2 15x

29. x3 2 8

x2 1 ax 2 2x 2 2a 30.

xy 1 2x 1 3y 1 6x3 1 27

Perform the operations and simplify, whenever possible. Assume that no denominators are 0.

31. x2 2 1

x?

x2

x2 1 2x 1 1

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A72 AppendixAAReviewofBasicAlgebra

32. y2 2 2y 1 1

y?

y 1 2y2 1 y 2 2

33. 3x2 1 7x 1 2

x2 1 2x?

x2 2 x3x2 1 x

34. x2 1 x

2x2 1 3x?

2x2 1 x 2 3x2 2 1

35. x2 1 xx 2 1

?x2 2 1x 1 2

36. x2 1 5x 1 6x2 1 6x 1 9

?x 1 2x2 2 4

37. 2x2 1 32

84

x2 1 162

38. x2 1 x 2 6x2 2 6x 1 9

4x2 2 4x2 2 9

39. z2 1 z 2 20

z2 2 44

z2 2 25z 2 5

40. ax 1 bx 1 a 1 b

a2 1 2ab 1 b2 4x2 2 1

x2 2 2x 1 1

41. 3x2 1 5x 2 2

x3 1 2x2 46x2 1 13x 2 5

2x3 1 5x2

42. x2 1 13x 1 128x2 2 6x 2 5

42x2 2 x 2 3

8x2 2 14x 1 5

43. x2 1 7x 1 12x3 2 x2 2 6x

?x2 2 3x 2 10x2 1 2x 2 3

?x3 2 4x2 1 3xx2 2 x 2 20

44. x 1x 2 22 2 3

x 1x 1 72 2 3 1x 2 12 ?x 1x 1 12 2 2

x 1x 2 72 1 3 1x 1 12

45. x2 2 2x 2 3

21x2 2 50x 2 16?

3x 2 8x 2 3

4x2 1 6x 1 5

7x2 2 33x 2 10

46. x3 1 27x2 2 4

4 ax2 1 4x 1 3x2 1 2x

4x2 1 x 2 6x2 2 3x 1 9

b

47. 3

x 1 31

x 1 2x 1 3

48. 3

x 1 11

x 1 2x 1 1

49. 4x

x 2 12

4x 2 1

50. 6x

x 2 22

3x 2 2

51. 2

5 2 x1

1x 2 5

52. 3

x 2 62

26 2 x

53. 3

x 1 11

2x 2 1

54. 3

x 1 41

xx 2 4

55. a 1 3

a2 1 7a 1 121

aa2 2 16

56. a

a2 1 a 2 21

2a2 2 5a 1 4

57. x

x2 2 42

1x 1 2

58. b2

b2 2 42

4b2 1 2b

59. 3x 2 2

x2 1 2x 1 12

xx2 2 1

60. 2t

t2 2 252

t 1 1t2 1 5t

61. 2

y2 2 11 3 1

1y 1 1

62. 2 14

t2 2 42

1t 2 2

63. 1

x 2 21

3x 1 2

23x 2 2x2 2 4

64. x

x 2 32

5x 1 3

13 13x 2 12

x2 2 9

65. a 1x 2 2

11

x 2 3b ?

x 2 32x

66. a 1x 1 1

21

x 2 2b 4

1x 2 2

67. 3x

x 2 42

xx 1 4

23x 1 116 2 x2

68. 7x

x 2 51

3x5 2 x

13x 2 1x2 2 25

69. 1

x2 1 3x 1 22

2x2 1 4x 1 3

11

x2 1 5x 1 6

70. 22

x 2 y1

2x 2 z

22z 2 2y

1y 2 x2 1z 2 x2

71. 3x 2 2

x2 1 x 2 202

4x2 1 2x2 2 25

13x2 2 25x2 2 16

72. 3x 1 2

8x2 2 10x 2 31

x 1 46x2 2 11x 1 3

21

4x 1 1

Simplify each complex fraction. Assume that no denomi-nators are 0.

73.

3ab

6acb2

74.

3t2

9xt

18x

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SectionA.6RationalExpressionsA73

75. 3a2bab27

76.

3u2v4t

3uv

77.

x 2 yab

y 2 xab

78.

x2 2 5x 1 62x2y

x2 2 92x2y

79.

1x

11y

xy 80.

xy11x

111y

81.

1x

11y

1x

21y

82.

1x

21y

1x

11y

83.

3ab

24a2

x1b

11

ax

84. 1 2

xy

x2

y2 2 1

85. x 1 1 2

6x

x 1 5 16x

86. 2z

1 23z

87. 3xy

1 21

xy

88. x 2 3 1

1x

21x

2 x 1 3

89. 3x

x 11x

90. 2x2 1 4

2 14x5

91.

xx 1 2

22

x 2 13

x 1 21

xx 2 1

92.

2xx 2 3

11

x 2 23

x 2 32

xx 2 2

Write each expression without using negative exponents, and simplify the resulting complex fraction. Assume that no denominators are 0.

93. 1

1 1 x21 94. y21

x21 1 y21

95. 3 1x 1 2221 1 2 1x 2 1221

1x 1 2221

96. 2x 1x 2 3221 2 3 1x 1 2221

1x 2 3221 1x 1 2221

applications 97. Engineering The stiffness k of the shaft shown in

the illustration is given by the following formula where k1 and k2 are the individual stiffnesses of each section. Simplify the complex fraction on the right side of the formula.

k 51

1k1

11k2

Section 1 Section 2

98. Electronics The combined resistance R of three resistors with resistances of R1, R2, and R3 is given by the following formula. Simplify the complex fraction on the right side of the formula.

R 51

1R1

11

R21

1R3

discovery and WritingSimplify each complex fraction. Assume that no denomi-nators are 0.

99. x

1 11

3x21

100. ab

2 13

2a21

101. 1

1 11

1 11x

102. y

2 12

2 12y

103. Explain why the formula ab

1cd

5ad 1 bc

bd is valid.

104. Explain why the formula ab

4cd

5ab

?dc

is valid.

105. Explain the Commutative Property of Addition and explain why it is useful.

106. Explain the Distributive Property and explain why it is useful.

review Write each expression without using absolute value sym-bols.

107. 026 0 108. 0 5 2 x 0 , given that x , 0

Simplify each expression.

109. ax3y22

x21yb

23

110. 127x62 2/3

111. "20 2 "45

112. 2 1x2 1 42 2 3 12x2 1 52 Not For Sale

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A74 AppendixAAReviewofBasicAlgebra

APPENDIX reVieW

SEctIoN A.1 SetsofRealNumbers

definitions and Concepts examples

Naturalnumbers: The numbers that we count with.

Wholenumbers: The natural numbers and 0.

Integers: The whole numbers and the negatives of the natural numbers.

Rationalnumbers: {x 0x can be written in the form ab 1b 2 02 , where a and b are integers.}

All decimals that either terminate or repeat.

Irrationalnumbers: Nonrational real numbers.All decimals that neither terminate nor repeat.

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, %

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, %

% , 25, 24, 23, 22, 21, 0, 1, 2, 3, 4, 5, %

2 521

, 25 5 251

, 23

, 273

, 0.25 514

34

5 0.75, 135

5 2.6, 23

5 0.666 % 5 0.6, 711

5 0.63

"2, 2"26, p, 0.232232223 %

Realnumbers: Any number that can be expressed as a decimal.

Primenumbers: A natural number greater than 1 that is divisible only by itself and 1.

Compositenumbers: A natural number greater than 1 that is not prime.

26, 2913

, 0, p, "31, 10.73

2, 3, 5, 7, 11, 13, 17, %

4, 6, 8, 9, 10, 12, 14, 15, %

Evenintegers: The integers that are exactly divisible by 2.

Oddintegers: The integers that are not exactly divisible by 2.

% , 26, 24, 22, 0, 2, 4, 6, %

% , 25, 23, 21, 1, 3, 5, %

AssociativeProperties: of addition 1a 1 b2 1 c 5 a 1 1b 1 c2 of multiplication 1ab2c 5 a 1bc2CommutativeProperties: of addition a 1 b 5 b 1 a

of multiplication ab 5 ba

DistributiveProperty: a 1b 1 c2 5 ab 1 ac

5 1 14 1 72 5 15 1 42 1 7

15 ? 42 ? 7 5 5 14 ? 72

4 1 7 5 7 1 4 1.7 1 2.5 5 2.5 1 1.7

4 ? 7 5 7 ? 4   1.7 12.52 5 2.5 11.72 3 1x 1 62 5 3x 1 3 ? 6 0.2 1y 2 102 5 0.2y 2 0.2 1102

DoubleNegativeRule: 2 12a2 5 a 2 1252 5 5   2 12a2 5 a

Openintervals have no endpoints.

Closedintervals have two endpoints.

Half-openintervals have one endpoint.

123, 22 )(–3 2

323, 2 4 ][–3 2

123, 2 4 ](–3 2

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AppendixReviewA75

definitions and Concepts examples

Absolutevalue:

If x $ 0, then 0 x 0 5 x.

If x , 0, then 0 x 0 5 2x.

0 7 0 5 7    0 3.5 0 5 3.5    ` 72` 5

72    0 0 0 5 0

027 0 5 7    023.5 0 5 3.5    `272` 5

72 

Distance:The distance between points a and b on a number line is d 5 0 b 2 a 0 .

The distance on the number line between points with coordinates of 23 and 2 is d 5 0 2 2 1232 0 5 0 5 0 5 5.

eXerCisesConsider the set 526,23,0,12,3,p,"5,6,86. List the numbers in this set that are 1. natural numbers.

2. whole numbers.

3. integers.

4. rational numbers.

5. irrational numbers.

6. real numbers.

Consider the set 526,23,0,12,3,p,"5,6,86. List the numbers in this set that are 7. prime numbers.

8. composite numbers.

9. even integers.

10. odd integers.

Determine which property of real numbers justifies each statement.

11. 1a 1 b2 1 2 5 a 1 1b 1 22

12. a 1 7 5 7 1 a

13. 4 12x2 5 14 ? 22x

14. 3 1a 1 b2 5 3a 1 3b

15. 15a27 5 7 15a2

16. 12x 1 y2 1 z 5 1y 1 2x2 1 z

17. 2 1262 5 6

Graph each subset of the real numbers: 18. the prime numbers between 10 and 20

19. the even integers from 6 to 14

Graph each interval on the number line. 20. 23 , x # 5

21. x $ 0 or x , 21

22. 122, 4 4

23. 12`, 22 d 125,`2

24. 12`,242 c 36,`2

Write each expression without absolute value symbols.

25. 0 6 0 26. 0225 0 27. 0 1 2 "2 0 28. 0 "3 2 1 0 29. On a number line, find the distance between points

with coordinates of 25 and 7.

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A76 AppendixAAReviewofBasicAlgebra

SEctIoN A.2 IntegerExponentsandScientificNotation

definitions and Concepts examples

Natural-numberexponents:

n factors of x

xn 5 x ? x ? x ? c ? x x5 5 x ? x ? x ? x ? x

Rulesofexponents:If there are no divisions by 0,

• xmxn 5 xm1n • 1xm2 n 5 xmn

• 1xy2 n 5 xnyn • axyb

n

5xn

yn

• x0 5 1 1x 2 02 • x2n 51xn

• xm

xn 5 xm2n • axyb

2n

5 ayxb

n

x2x5 5 x215 5 x7 1x22 7 5 x2?7 5 x14

1xy2 5 5 x5y5 axyb

5

5x5

y5

60 5 1 622 5162 5

136

x6

x4 5 x624 5 x2 ax6b

23

5 a6xb

3

563

x3 5216x3

Scientificnotation:A number is written in scientificnotation when it is writ-ten in the form N 3 10n, where 1 # 0N 0 , 10.

Write each number in scientific notation.386,000 5 3.86 3 105 0.0025 5 2.5 3 1023

Write each number in standard form.7.3 3 103 5 7,300 5.25 3 1024 5 0.000525

eXerCisesWrite each expression without using exponents.

30. 25a3 31. 125a2 2

Write each expression using exponents.

32. 3ttt 33. 122b2 13b2

Simplify each expression.

34. n2n4 35. 1p32 2

36. 1x3y22 4 37. aa4

b2b3

38. 1m23n02 2 39. ap22q2

2b

3

40.a5

a8 41. aa2

b3b22

42. a3x2y22

x2y2 b22

43. aa23b2

ab23 b22

44. a23x3yxy3 b

22

45. a22m22n0

4m2n21b23

46. If x 5 23 and y 5 3, evaluate 2x2 2 xy2

Write each number in scientific notation.

47. 6,750 48. 0.00023

Write each number in standard notation.

49. 4.8 3 102 50. 0.25 3 1023

51. Use scientific notation to simplify 145,0002 1350,0002

0.000105.

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AppendixReviewA77

SEctIoN A.3 RationalExponentsandRadicals

definitions and Concepts examples

Summaryof a1/n definitions:• If a $ 0, then a1/n is the nonnegative number b

such that bn 5 a.• If a , 0 and n is odd, then a1/n is the real number

b such that bn 5 a.• If a , 0 and n is even, then a1/n is not a real

number.

161/2 5 4 because 42 5 16

12272 1/3 5 23 because 1232 3 5 227

12162 1/2 is not a real number because no real number squared is 216.

Ruleforrationalexponents:If m and n are positive integers, m

n is in lowest terms, and a1/n is a real number, then

am/n 5 1a1/n2m 5 1am2 1/n

82/3 5 181/32 2 5 22 5 4 or 1822 1/3 5 641/3 5 4

Definitionof "n a:

"na 5 a1/n "3 125 5 1251/3 5 5

Propertiesofradicals:If all radicals are real numbers and there are no divi-sions by 0, then

• "nab 5 "n

a"nb

• Ån a

b5

"na

"nb

• m#"n

a 5 #n "m a 5mn"a

"5 32x10 5 "5 32"5 x10 5 2x2

Å4 x12

6255

"4 x12

"4 6255

x12/4

55

x3

5

#3 "64 5 "3 8 5 2  #"3 64 5 "4 5 2 2 # 3"64 5 "6 64 5 2

eXerCisesSimplify each expression, if possible.

52. 1211/2 53. a 27125

b1/3

54. 132x52 1/5 55. 181a42 1/4

56. 121,000x62 1/3 57. 1225x22 1/2

58. 1x12y22 1/2 59. ax12

y4 b21/2

60. a2c2/3c5/3

c22/3 b1/3

61. aa21/4a3/4

a9/2 b21/2

Simplify each expression.

62. 642/3 63. 3223/5

64. a1681

b3/4

65. a 32243

b2/5

66. a 827

b22/3

67. a 16625

b23/4

68. 12216x32 2/3 69. pa/2pa/3

pa/6

Simplify each expression.

70. "36 71. 2"49

72. Å925

73. Å3 27

125

74. "x2y4 75. "3 x3

76. Å4 m8n4

p16 77. Å5 a15b10

c5

Simplify and combine terms.

78. "50 1 "8 79. "12 1 "3 2 "27

80. "3 24x4 2 "3 3x4

Rationalize each denominator.

81."7

"5 82.

8

"8

83.1

"3 2 84.

2

"3 25

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A78 AppendixAAReviewofBasicAlgebra

Rationalize each numerator.

85."2

5 86.

"55

87."2x

3 88.

3"3 7x2

SEctIoN A.4 Polynomials

definitions and Concepts examples

Monomial: A polynomial with one term.

Binomial: A polynomial with two terms.

Trinomial: A polynomial with three terms.

2, 3x, 4x2y, 2x3y2z

2t 1 3, 3r2 2 6r, 4m 1 5n

3p2 2 7p 1 8, 3m2 1 2n 2 p

The degreeofamonomial is the sum of the exponents on its variables.

The degreeofapolynomial is the degree of the term in the polynomial with highest degree.

The degree of 4x2y3 is 2 1 3 5 5

The degree of the first term of 3p2q3 2 6p3q4 1 9pq2 is 2 1 3 5 5, the degree of the second term is 3 1 4 5 7, and the degree of the third term is 1 1 2 5 3. The degree of the polynomial is the largest of these. It is 7.

Multiplyingamonomialtimesapolynomial:

a 1b 1 c 1 d 1 c 2 5 ab 1 ac 1 ad 1 c 3 1x 1 22 5 3x 1 62x 13x2 2 2y 1 32 5 6x3 2 4xy 1 6x

Additionandsubtractionofpolynomials:To add or subtract polynomials, removeparentheses and combine like terms.

Add: 13x2 1 5x2 1 12x2 2 2x2 . 13x2 1 5x2 1 12x2 2 2x2 5 3x2 1 5x 1 2x2 2 2x

5 3x2 1 2x2 1 5x 2 2x

5 5x2 1 3x

Subtract: 14a2 2 5b2 2 13a2 2 7b2 . 14a2 2 5b2 2 13a2 2 7b2 5 4a2 2 5b 2 3a2 1 7b

5 4a2 2 3a2 2 5b 1 7b

5 a2 1 2b

Specialproducts:

1x 1 y2 2 5 x2 1 2xy 1 y2

1x 2 y2 2 5 x2 2 2xy 1 y2

1x 1 y2 1x 2 y2 5 x2 2 y2

12m 1 32 2 5 4m2 1 12m 1 9

14t 2 3s2 2 5 16t2 2 24ts 1 9s2

12m 1 n2 12m 2 n2 5 4m2 2 2mn 1 2mn 2 n2 5 4m2 2 n2

Multiplyingabinomialtimesabinomial:Use the FOIL method. 12a 1 b2 1a 2 b2 5 2a 1a2 1 2a 12b2 1 ba 1 b 12b2

5 2a2 2 2ab 1 ab 2 b2

5 2a2 2 ab 2 b2

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AppendixReviewA79

definitions and Concepts examples

The conjugate of a 1 b is a 2 b.

To rationalize the denominator of a radical expression, multiply both the numerator and denominator of the rational expression by the conjugate of the denominator.

Rationalize the denominator: x

"x 1 2.

x

"x 1 25

x 1"x 2 221"x 1 22 1"x 2 22

Multiply the numera-

tor and denominator

by the conjugate of

"x 1 2.

5x"x 2 2x

x 2 4 Simplify.

Divisionofpolynomials:To divide polynomials, use long division.

Divide: 2x 1 3q6x3 1 7x2 2 x 1 3.

3x2 2 x 1 1 2x 1 3q 6x3 1 7x2 2 x 1 3

6x3 1 9x2

2 2x2 2 x2 2x2 2 3x

1 2x 1 31 2x 1 3

0

eXerCisesGive the degree of each polynomial and tell whether the polynomial is a monomial, a binomial, or a trinomial.

89. x3 2 8

90. 8x 2 8x2 2 8

91."3x2

92. 4x4 2 12x2 1 1

Perform the operations and simplify. 93. 2 1x 1 32 1 3 1x 2 42 94. 3x2 1x 2 12 2 2x 1x 1 32 2 x2 1x 1 22 95. 13x 1 22 13x 1 22 96. 13x 1 y2 12x 2 3y2 97. 14a 1 2b2 12a 2 3b2 98. 1z 1 32 13z2 1 z 2 12 99. 1an 1 22 1an 2 12 100. Q"2 1 xR2

101. Q"2 1 1RQ"3 1 1R 102. Q"3 3 2 2RQ"3 9 1 2"3 3 1 4R

Rationalize each denominator.

103.2

"3 2 1 104.

22

"3 2 "2

105.2x

"x 2 2 106.

"x 2 "y

"x 1 "y

Rationalize each numerator.

107."x 1 2

5 108.

1 2 "aa

Perform each division.

109.3x2y2

6x3y 110.

4a2b3 1 6ab4

2b2

111. 2x 1 3q2x3 1 7x2 1 8x 1 3

112. x2 2 1qx5 1 x3 2 2x 2 3x2 2 3

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A80 AppendixAAReviewofBasicAlgebra

SEctIoN A.5 FactoringPolynomials

definitions and Concepts examples

Factoringoutacommonmonomial:

ab 1 ac 5 a 1b 1 c2 3p3 2 6p2q 1 9p 5 3p 1p2 2 2pq 1 32Factoringthedifferenceoftwosquares:

x2 2 y2 5 1x 1 y2 1x 2 y2 4x2 2 9 5 12x 1 32 12x 2 32Factoringtrinomials:

• Trinomial squares

x2 1 2xy 1 y2 5 1x 1 y2 2

x2 2 2xy 1 y2 5 1x 2 y2 2

• To factor general trinomials use trial and error or grouping.

9a2 1 12ab 1 4b2 5 13a 1 2b2 13a 1 2b2 5 13a 1 2b2 2

r2 2 4rs 1 4s2 5 1r 2 2s2 1r 2 2s2 5 1r 2 2s2 2

6x2 2 5x 2 6 5 12x 2 32 13x 1 22

Factoringthesumanddifferenceoftwocubes:

x3 1 y3 5 1x 1 y2 1x2 2 xy 1 y22x3 2 y3 5 1x 2 y2 1x2 1 xy 1 y22

r3 1 8 5 1r 1 22 1r2 2 2r 1 4227a3 2 8b3 5 13a 2 2b2 19a2 1 6ab 1 4b22

eXerCisesFactor each expression completely, if possible.

113. 3t3 2 3t 114. 5r3 2 5

115. 6x2 1 7x 2 24 116. 3a2 1 ax 2 3a 2 x

117. 8x3 2 125 118. 6x2 2 20x 2 16

119. x2 1 6x 1 9 2 t2 120. 3x2 2 1 1 5x

121. 8z3 1 343 122. 1 1 14b 1 49b2

123. 121z2 1 4 2 44z 124. 64y3 2 1,000

125. 2xy 2 4zx 2 wy 1 2zw 126. x8 1 x4 1 1

SEctIoN A.6 AlgebraicFractions

definitions and Concepts examples

Propertiesoffractions:If there are no divisions by 0, then

• ab

5cd

if and only if ad 5 bc.

• ab

5axbx

• ab

?cd

5acbd

• ab

4cd

5adbc

3x4

56x8

because 13x28 and 4 16x2 both equal 24x.

6x2

8x3 53 ? 2 ? x ? x

2 ? 4 ? x ? x ? x5

34x

3p2q

?2p6q

53p ? 2p2q ? 6q

53p ? 2p

2q ? 3 ? 2q5

p2

2q2

2t3s

42t6s

52t3s

?6s2t

52t ? 6s3s ? 2t

52t ? 2 ? 3s

3s2t5 2

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AppendixReviewA81

definitions and Concepts examples

• ab

1cb

5a 1 c

b •

ab

2cb

5a 2 c

b

• a ? 1 5 a • a1

5 a • aa

5 1

• ab

52a2b

5 2a

2b5 2

2ab

• 2ab

5a

2b5

2ab

5 22a2b

x4

1y4

5x 1 y

4

3p2q

2p2q

53p 2 p

2q5

2p2q

5pq

7 ? 1 5 7 71

5 7 77

5 1

72

52722

5 27

225 2

272

272

57

225

272

5 22722

Simplifyingrationalexpressions:To simplify a rational expression, factor the numerator and denominator, if possible, and divide out factors that are common to the numerator and denominator.

To simplify 2 2 x2x 2 4, factor 21 from the numerator and

2 from the denominator to get

2 2 x2x 2 4

521 122 1 x2

2 1x 2 22 521 1x 2 222 1x 2 22 5

212

5 212

Addingorsubtractingrationalexpressions:To add or subtract rational expressions with unlike denominators, find the LCD of the expressions, write each expression with a denominator that is the LCD, add or subtract the expressions, and simplify the result, if possible.

2x

x 1 22

2xx 2 3

52x 1x 2 32

1x 1 22 1x 2 32 22x 1x 1 22

1x 2 32 1x 1 22

52x 1x 2 32 2 2x 1x 1 22

1x 1 22 1x 2 32

52x2 2 6x 2 2x2 2 4x

1x 1 22 1x 2 32

5210x

1x 1 22 1x 2 32Complexfractions:To simplify complex fractions, multiply the numerator and denominator of the complex fraction by the LCD of all the fractions.

1 2y2

1y

112

5

2ya1 2y2b

2ya1y

112b

52y 2 y2

2 1 y5

y 12 2 y22 1 y

eXerCisesSimplify each rational expression.

127. 2 2 x

x2 2 4x 1 4 128.

a2 2 9a2 2 6a 1 9

Perform each operation and simplify. Assume that no denominators are 0.

129.x2 2 4x 1 4

x 1 2?

x2 1 5x 1 6x 2 2

130.2y2 2 11y 1 15

y2 2 6y 1 8?

y2 2 2y 2 8y2 2 y 2 6

131.2t2 1 t 2 33t2 2 7t 1 4

410t 1 15

3t2 2 t 2 4

132.p2 1 7p 1 12p3 1 8p2 1 4p

4p2 2 9

p2

133.x2 1 x 2 6x2 2 x 2 6

?x2 2 x 2 6x2 1 x 2 2

4x2 2 4

x2 2 5x 1 6

134. a2x 1 6x 1 5

42x2 2 2x 2 4

x2 2 25b x2 2 x 2 2

x2 2 2x 2 15

135.2

x 2 41

3xx 1 5

136.5x

x 2 22

3x 1 7x 1 2

12x 1 1x 1 2

137.x

x 2 11

xx 2 2

1x

x 2 3

138.x

x 1 12

3x 1 7x 1 2

12x 1 1x 1 2

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A82 AppendixAAReviewofBasicAlgebra

139.3 1x 1 12

x2

5 1x2 1 32x2 1

xx 1 1

140.3x

x 1 11

x2 1 4x 1 3x2 1 3x 1 2

2x2 1 x 2 6

x2 2 4

Simplify each complex fraction. Assume that no denomi-nators are 0.

141.

5x2

3x2

8

142.

3xy

6xy2

143.

1x

11y

x 2 y 144.

x21 1 y21

y21 2 x21

APPENDIX test

Consider the set 527,223,0,1,3,"10,46.

1. List the numbers in the set that are odd integers.

2. List the numbers in the set that are prime numbers.

Determine which property justifies each statement.

3. 1a 1 b2 1 c 5 1b 1 a2 1 c

4. a 1b 1 c2 5 ab 1 ac

Graph each interval on a number line.

5. 24 , x # 2 6. 12`, 232 c 36,`2

Write each expression without using absolute value sym-bols.

7. 0217 0 8. 0 x 2 7 0 , when x , 0

Find the distance on a number line between points with the following coordinates. 9. 24 and 12 10. 220 and 212

Simplify each expression. Assume that all variables repre-sent positive numbers, and write all answers without using negative exponents.

11. x4x5x2 12. r2r3sr4s2

13.1a21a2222

a23 14. ax0x2

x22 b6

Write each number in scientific notation.

15. 450,000 16. 0.000345

Write each number in standard notation.

17. 3.7 3 103 18. 1.2 3 1023

Simplify each expression. Assume that all variables repre-sent positive numbers, and write all answers without using negative exponents.

19. 125a42 1/2 20. a3681

b3/2

21. a 8t6

27s9b22/3

22. "3 27a6

23. "12 1 "27 24. 2"3 3x4 2 3x"3 24x

25. Rationalize the denominator: x

"x 2 2.

26. Rationalize the numerator: "x 2 "y

"x 1 "y.

Perform each operation.

27. 1a2 1 32 2 12a2 2 42 28. 13a3b22 122a3b42 29. 13x 2 42 12x 1 72 30. 1an 1 22 1an 2 32 31. 1x2 1 42 1x2 2 42 32. 1x2 2 x 1 22 12x 2 32 33. x 2 3q6x2 1 x 2 23

34. 2x 2 1q2x3 1 3x2 2 1

Factor each polynomial.

35. 3x 1 6y

36. x2 2 100

37. 10t2 2 19tw 1 6w2

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AppendixTestA83

38. 3a3 2 648

39. x4 2 x2 2 12

40. 6x4 1 11x2 2 10

Perform each operation and simplify if possible. Assume that no denominators are 0.

41.x

x 1 21

2x 1 2

42.x

x 1 12

xx 2 1

43.x2 1 x 2 20

x2 2 16?

x2 2 25x 2 5

44.x 1 2

x2 1 2x 1 14

x2 2 4x 1 1

Simplify each complex fraction. Assume that no denomi-nators are 0.

45.

1a

11b

1b

46. x21

x21 1 y21

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