Section 9.1- Radical Expressions and Graphs · ... rational, irrational, or not a real number ... distance between two points in the coordinate plane is derived from the Pythagorean

1 | P a g e

Hannah Province – Mathematics Department – Southwest Tennessee Community College

Chapter 9

Section 9.1- Radical Expressions and Graphs Objective:

1. Find square roots. 2. Decide whether a given root is rational, irrational, or not a

real number. 3. Find cube, fourth, and other roots. 4. Use a calculator to find roots.

Find square roots. Find all square roots of 49.

To find the square roots of 49, think of a number than when

multiplied by itself gives 49. On square root is 7 because 7 7 =

49. Another square root of 49 is 7 because (7)(7) = 49. The

number 49 has two square roots, 7 and 7; one is positive and one is negative.

The positive or principal square root of a number is written with

the symbol √ For example, the positive square root of 121 is 11, written

The -√ is used for the negative square root of a number. For

example, the negative square root of 121 is 11, written The symbol is called a radical sign, always represents the positive square root.

2 | P a g e


The number inside the radical sign is called the radicand, and the entire expression, radical sign and radicand, is called a radical.

An algebraic expression containing a radical is called a radical expression.

We know that 422 so the reverse of this process the

24 .

981 since 8192

11 since 112

864 since 6482

3 | P a g e


Ex. Find each square root.

22

169

36

9

49

4

12

180

Ex. Find the square of each radical expression

√

-√

√

4 | P a g e


Decide whether a given root is rational, irrational, or not a real number. All numbers with square roots that are rational are called perfect squares.

Ex. Tell whether each square root is rational, irrational, or not a real number.

√

√

√

5 | P a g e


Find cube, fourth, and other roots Finding the square root of a number is the inverse (reverse) of squaring a number. There are inverses to finding the cube of a number, or finding the fourth or higher power of a number. These inverses are the cube root and fourth root.

Ex. Find each cube root.

√

√

√

6 | P a g e


When a radical has an even index (square root, fourth root, and so on), the radicand must be nonnegative to yield a real number root.

Ex. Find each root.

√

√

√

√

√

7 | P a g e


Use a calculator to find roots. Radical expressions often represent irrational numbers. To find approximations of such radicals, we usually use a calculator. For

example,

where the symbol means “is approximately equal to.” Ex. Use a calculator to verify that each approximation is correct.

Section 9.2 – Rational Expressions

8 | P a g e


Objectives

1. Use exponential notation for nth roots. 2. Define and use expressions of the form a

m/n.

3. Convert between radicals and rational exponents. 4. Use the rules for exponents with rational exponents.

Rational Exponents

nn xx

xx

xx

xx

xx

1

51

5

41

4

31

3

21

Recall rules

n can be any index when a is nonnegative

n must be odd when a is negative

9 | P a g e


Ex: =

=

=

- =

What about ?

10 | P a g e


CAUTION Notice the difference between parts (c) and (d) in Example 1. The radical in part (c) is the negative fourth root of a positive number, while the radical in part (d) is the principal fourth root of a negative number, which is not a real number. Definition of a

m/n

11 | P a g e


Definition: If a is a real number, m is an integer, and n is a positive integer greater than 1, then

mnn mn

m

aaa where m is the power you are

raising and n is the root you are taking. (Note: We assume that a is nonnegative if n is even.) Recall the Properties of Exponents that also hold true for rational

numbers as exponents:

For nonzero real numbers a and b and rational numbers m and n ,

1. 1aa ( a is any real number.)

2. 10 a

3. nmnm aaa

4. nm

n

m

aa

a

5. nmnm aa

6. n

n

aa

1

and n

na

a

1

7. nnnbaba

8. n

nn

b

a

b

a

In order for an exponential expression involving products and/or quotients to be SIMPLIFIED . . .

1. Each base must occur only once. 2. Each base must be raised to a single power. 3. All the powers must be positive.

12 | P a g e


Ex: =

(

)

=

=

=

13 | P a g e


(

)

=

(

)

=

(

)=

14 | P a g e


Alternative Definition of am/n

Converting between Rational Exponents and Radicals Ex: 15

1/2

4n

2/3

15 | P a g e


7h3/4

– (2h)2/5

g–4/5

10

5/6

16 | P a g e


Applying Rules for Rational Exponents Write with only positive exponents. Assume that all variables represent positive real numbers. Do not make the common mistake of multiplying exponents in the first step. 6

3/4 · 6

1/2

17 | P a g e


Applying Rules for Rational Exponents Rewrite all radicals as exponentials, and then apply the rules for rational exponents. Leave answers in exponential form. Assume that all variables represent positive real numbers.

18 | P a g e


Section 9.3 – Simplifying Rational Expressions Objectives:

1. Use the product rule for radicals. 2. Use the quotient rule for radicals. 3. Simplify radicals. 4. Use the Pythagorean formula. 5. Use the distance formula.

Use the Product Rule for Radicals

Ex: √ √

19 | P a g e


√ √

√

√

√ √

20 | P a g e


Use the Quotient Rule for Radicals

√

√

√

√

21 | P a g e


Simplifying Radicals

√

√

√

√

22 | P a g e


Simplifying Radicals with Variables

√

√

√

√

23 | P a g e


Pythagorean Formula The Pythagorean formula relates lengths of the sides of a right triangle

Ex:

24 | P a g e


The Distance Formula

The distance formula, which allows us to compute the distance between two points in the coordinate plane is derived from the Pythagorean formula. Ex: Find the distance between (1, 6) and (4, –2).

25 | P a g e


Section 9.7- Complex Numbers Objective:

1. Simplifying numbers of the form √ , where b >0. Introduction to Complex Numbers

Radicals such as 4 is not a real number because no real number squared produces a negative number.

In general, n x for any even n and –x is not a real number.

However, we can simplify using complex numbers.

1i and 112

2 i

26 | P a g e


Examples

Ex. 25

36

24

45

27 | P a g e


The standard form of a complex number is a + bi. Where ‘a’ is the real part and ‘b’ is the imaginary part.

Identify the real and imaginary part of each complex number.

4 – 2i

-3i

5

28 | P a g e


Adding and Subtracting Complex Numbers To add/subtract complex numbers you combine the real parts together and then combine the imaginary parts together.

Ex. (-5 – 7i) + (-2 – 2i)=

Ex. iii 586442 =

29 | P a g e


Ex. 818646 =

Ex. 17253 =

30 | P a g e


Multiplying Square Roots of Negative Numbers

Ex. √ √ =

Ex. √ √ =

31 | P a g e


Dividing Square Roots of Negative Numbers

Ex. √

√ =

Ex. √

√ =

Documents

Section 9.1- Radical Expressions and Graphs · ... rational, irrational, or not a real number ... distance between two points in the coordinate plane is derived from the Pythagorean