chapter 8 section 1 copyright © 2008 pearson education, inc. publishing as pearson addison-wesley

Chapter Chapter 88Section Section 11

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley


Evaluating Roots

Find square roots.Decide whether a given root is rational, irrational, or not a real number.Find decimal approximations for irrational square roots.Use the Pythagorean formula.Use the distance formula.Find cube, fourth, and other roots.

11

44

33

22

66

55

8.18.18.18.1


Objective 11

Find square roots.

Slide 8.1 - 3


Find square roots.

When squaring a number, multiply the number by itself. To find the square root of a number, find a number that when multiplied by itself, results in the given number. The number a is called a square root of the number a

2.

Slide 8.1 - 4


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

the symbol .

Find square roots. (cont’d)

Slide 8.1 - 5

0 0

a

Radical Sign Radicand

The symbol , is called a radical sign, always represents the

positive square root (except that ). The number inside the

radical sign is called the radicand, and the entire expression—radical

sign and radicand—is called a radical.

The symbol – is used for the negative square root of a number.


Find square roots. (cont’d)

Slide 8.1 - 6

The statement is incorrect. It says, in part, that a positive number equals a negative number.

9 3


EXAMPLE 1

Find all square roots of 64.

Solution:

Finding All Square Roots of a Number

Slide 8.1 - 7

Positive Square Root

Negative Square Root

64 8

64 8


Find each square root.

EXAMPLE 2

Solution:

Finding Square Roots

Slide 8.1 - 8

169

225

13

15

25

64

25

64 5

8


EXAMPLE 3 Squaring Radical Expressions

Slide 8.1 - 9

Find the square of each radical expression.

Solution:

17 2

17 17

29 2

29 29

22 3x 222 3x 22 3x


Objective 22

Decide whether a given root is rational, irrational, or not a real number.

Slide 8.1 - 10


Deciding whether a given root is rational, irrational, or not a real number.

Slide 8.1 - 11

All numbers with square roots that are rational are called perfect squares.

Perfect Squares Rational Square Roots

25

144

4

9

25 5

144 12

4 2

9 3

A number that is not a perfect square has a square root that is irrational. Many square roots of integers are irrational.

Not every number has a real number square root. The square of a real number can never be negative. Therefore, is not a real number.

-36


EXAMPLE 4Identifying Types of Square Roots

Slide 8.1 - 12

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

27 irrational

36 26 rational

27 not a real number

Solution:

Not all irrational numbers are square roots of integers. For example (approx. 3.14159) is a irrational number that is not an square root of an integer.


Objective 33

Slide 8.1 - 13

Find decimal approximations for irrational square roots.


Find decimal approximations for irrational square roots.

Slide 8.1 - 14

A calculator can be used to find a decimal approximation even if a number is irrational.


EXAMPLE 5 Approximating Irrational Square Roots

Slide 8.1 - 15

Find a decimal approximation for each square root. Round answers to the nearest thousandth.

Solution:

190 13.784048 13.784

99 9.9498743 9.950


Objective 44

Slide 8.1 - 16

Use the Pythagorean formula.


Many applications of square roots require the use of the Pythagorean formula.

If c is the length of the hypotenuse of a right triangle, and a and b are the lengths of the two legs, then

Slide 8.1 - 17

Use the Pythagorean formula.

2 2 2.a b c

Be careful not to make the common mistake thinking that

equals .

2 2a b

a b


EXAMPLE 6

2 2 213 15a 2 169 225a

Using the Pythagorean Formula

Slide 8.1 - 18

7, 24a b

Find the length of the unknown side in each right triangle.

Give any decimal approximations to the nearest thousandth.

15, 13c b

118

?

2 2 27 24 c 249 576 c 2625 c

625c 252 56a

56a 7.483

2 2 28 11b 264 121b 2 57b

57b 7.550

Solution:


EXAMPLE 7 Using the Pythagorean Formula to Solve an Application

Slide 8.1 - 19

A rectangle has dimensions of 5 ft by 12 ft. Find the length

of its diagonal.

5 ft

12 ft

Solution:

2 2 25 12 c 225 144 c

2169 c

169c

13ftc


Objective 55

Use the distance formula.

Slide 8.1 - 20

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8.1 - 21

Use the distance formula.

The distance between the points and is

1 1,x y 2 2,x y

2 2

2 1 2 1 .d x x y y


EXAMPLE 8 Using the Distance Formula

Slide 8.1 - 22

Find the distance between and . 6,3 2, 4

2 22 6 4 3d

Solution:

224 7d

65d

16 49d


Objective 66

Find cube, fourth, and other roots.

Slide 8.1 - 23


Finding the square root of a number is the inverse of squaring a

number. In a similar way, there are inverses to finding the cube

of a number or to finding the fourth or greater power of a

number.

The nth root of a is written

Find cube, fourth, and other roots.

.n a

Slide 8.1 - 24

n a

n a

Radical signIndex

Radicand

In , the number n is the index or order of the radical.

It can be helpful to complete and keep a list to refer to of third and fourth powers from 1-10.


Find each cube root. Solution:

EXAMPLE 9 Finding Cube Roots

Slide 8.1 - 25

3 64

3 27

3 512

4

3

8


EXAMPLE 10 Finding Other Roots

Slide 8.1 - 26

Find each root.

4 81

4 81

4 81

5 243

5 243

3

3

Not a real number.

3

3

Solution:

chapter 8 section 1 copyright © 2008 pearson education, inc. publishing as pearson addison-wesley

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irrational square roots

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types of square roots

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