Warm UpSimplify each expression.

1. 62

36 2. 112 121

3. (–9)(–9) 81 4. 2536

Write each fraction as a decimal.

5. 25

596.

7. 5 38

8. –1 56

0.4

5.375

0.5

–1.83

Roots and Roots and Irrational Irrational NumbersNumbers

Section 1.5

Objectives:

In this lesson you’ll:

• Evaluate expressions containing roots. • Classify numbers within the real number system

Words to know…

• Square root Square root - a number which, when multiplied by itself, produces the given number. (Ex. 7² = 49, 7 is the square

• root of 49)

• Perfect square-Perfect square- any number that has an integer square root.(ex. 100 is a perfect square ,

• Cube root Cube root - a number that is raised to the third power to form a product is a cube root. (ex 23=8, =2)

10100

Square RootsSquares

0² = 0

1² = 1

2² = 4

3² = 9

4² = 16

5² = 25

6² = 36

7² = 49

8² = 64

9² = 81

10² = 100

Perfect Square Roots

00

11 24 39 416

525

636 749

864

Are squares and square roots inverses?

932

2552

8192

39

525

981

A square root is the inverse operation of a square!

Do you know your perfect squares?

1) 49 ?

2) 64 ?

3) 9 ?

5)112 ?

4)52 ?

6)142 ?

7 and -7

8 and -8

3 and -3

25

121

196

Square RootsPositive real numbers have two square roots.

Find the square roots of 16.

The square roots of 16 are 4 and - 4.

4 4 = 42 = 16 = 4 Positive squareroot of 16

(–4)(–4) = (–4)2 = 16 = –4 Negative squareroot of 16

–

Writing Math

The small number to the left of the root is the index. In a square root, the index is understood to be 2. In other words, is the same as .

A number that is raised to the third power to form a product is a cube root of that product. The symbol indicates a cube root. Since 23 = 8, = 2. Similarly, the symbol indicates a fourth root: 2 = 16, so = 2.

Cube roots

Find each root.

Think: What number squared equals 81?

Think: What number squared equals 25?

You try

Think: What number cubed equals –216?

C.

= –6 (–6)(–6)(–6) = 36(–6) = –216

Finding Roots of Fractions.You try

Think: What number squared equals

a.

Think: What number cubed equals

b.

Finding Roots of Fractions.You try

Think: What number squared

equals

A.

Think: What number cubed equals

B.

Square roots of numbers that are not perfect squares, such as 15, are not whole numbers. A calculator can approximate the value of as 3.872983346... Without a calculator, you can use square roots of perfect squares to help estimate the square roots of other numbers.

If a whole number is not a perfect square, then its square root is irrational. For example, 2 is not a perfect square and is irrational.

Remember

Approximating Square Roots

Approximating Square Roots

Approximate to the nearest whole number.54

4972 6482

54 Is between 7² and 8². 3.754

Let’s practice…

8

17

24

Determine what two consecutive integers each root lies between.

35

Between 2 and 3

Between 4 and 5

Between 4 and 5

Between 5 and 6

Words to know…• Natural numbers Natural numbers - The counting numbers. (example: 1, 2,

3…)

• Whole numbers Whole numbers - The natural numbers and zero.(example: 0, 1,2,3…)

• Integers Integers -The whole numbers and their opposites.(ex: …-3,-2,-1,0,1,2,3…)

• Rational numbers Rational numbers - Numbers that can be expressed as a fraction (a/b).

Words to know…

• Terminating decimal Terminating decimal -Rational numbers in decimal form that have finite (ends) number of digits. (ex 2/5= 0.40 )

• Repeating decimal Repeating decimal -rational numbers in decimal form that have a block for one or more digits that repeats continuously. (ex. 1.3=1.333333333)

• Irrational numbers Irrational numbers - numbers that cannot be expressed as a fraction including square roots of whole numbers that are not perfect squares and nonterminating decimals that do not repeat.

The real numbers are made up of all rational and irrational numbers.

Note the symbols for the sets of numbers.R: real numbersQ: rational numbersZ: integersW: whole numbersN: natural numbers

Reading Math

Classifying Real Numbers

Write all classifications that apply to each real number.

A.

–32 = –

32 1

rational number, integer, terminating decimal

B.

irrational

–32

–32 can be written in the form .

14 is not a perfect square, so is irrational.

–32 can be written as a terminating decimal.

–32 = –32.0

Write all classifications that apply to each real number.

a. 7

rational number, repeating decimal

Check It Out!

67 9 = 7.444… = 7.4

7 can be written in the form .4 9

can be written as a repeating decimal.

b. –12 –12 can be written in the form .

–12 can be written as a terminating decimal.

rational number, terminating decimal, integer

Write all classifications that apply to each real number.

irrational

100 is a perfect square, so is rational.

10 is not a perfect square, so is irrational.

10 can be written in the form and as a terminating decimal.

natural, rational, terminating decimal, whole, integer

A challenge…• Would you know how to solve this….

36112 x

252 x252 x

A challenge…• Solve the variable.

6732 x

642 x

642 x

Find each square root.

1. 2.

3. 4.3

5. The area of a square piece of cloth is 68 in2. Estimate to the nearest tenth the side length of the cloth. 8.2 in.

Lesson Quiz

Write all classifications that apply to each real number.

6. –3.89 7.rational, repeating decimal

irrational

15