Chapter 8

Chapter 8 Section 1

Objectives

1

Copyright © 2012, 2008, 2004 Pearson Education, Inc.

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 theorem.

Use the distance formula.

Find cube, fourth, and other roots.

8.1

2

3

4

5

6


Objective 1

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

Square Root

A number b is a square root of a if b2 = a.


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 positive or principal square root of a number is written with

the symbol .

0 0

a

Radical SignRadicand

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

Slide 8.1-5

Find square roots. (cont’d)


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

9 3

Slide 8.1-6

Find square roots. (cont’d)


Find all square roots of 64.

Solution:

Positive Square Root

Negative Square Root

64 8

64 8

Slide 8.1-7

EXAMPLE 1 Finding All Square Roots of a Number


Find each square root.

Solution:

169

225

13

15

25

64

25

64 5

8

Slide 8.1-8

EXAMPLE 2 Finding Square Roots


Find the square of each radical expression.

Solution:

17 2

17 17

31 2

31 31

22 3x 222 3x 22 3x

Slide 8.1-9

EXAMPLE 3 Squaring Radical Expressions


Objective 2

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.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

Slide 8.1-11


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.

Slide 8.1-12

EXAMPLE 4 Identifying Types of Square Roots


Objective 3


Slide 8.1-13



Even if a number is irrational, a decimal that approximates the number can be found using a calculator.

Slide 8.1-14


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

Slide 8.1-15

EXAMPLE 5 Approximating Irrational Square Roots


Objective 4


Slide 8.1-16


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


2 2 2.a b c

Be careful not to make the common mistake thinking that

equals

2 2a b.a b

Slide 8.1-17


2 2 213 15a 2 169 225a

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:

Slide 8.1-18

EXAMPLE 6 Using the Pythagorean Theorem


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

Slide 8.1-19

EXAMPLE 7 Using the Pythagorean Theorem to Solve an Application


Objective 5


Slide 8.1-20



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

Slide 8.1-21


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

2 22 6 4 3d

Solution:

224 7d

65d

16 49d

Slide 8.1-22

EXAMPLE 8 Using the Distance Formula


Objective 6


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


.n a

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

n a

Radical sign

IndexRadicand

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

Slide 8.1-24


Find each cube root.

3 64

3 27

3 512

4

3

8

Slide 8.1-25

EXAMPLE 9 Finding Cube Roots

Solution:


Find each root.

4 81

4 81

4 81

5 243

5 243

3

3

Not a real number.

3

3

Solution:

Slide 8.1-26

EXAMPLE 10 Finding Other Roots

Chapter 8

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