10.1 Radical Expressions and Graphs

Objective 1

Find square roots.

Slide 10.1-3

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 10.1-4

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—radicalsign 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 10.1-5

Find square roots. (cont’d)

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

9 3

Slide 10.1-6

Find square roots. (cont’d)

Find all square roots of 64.

Solution:

Positive Square Root

Negative Square Root

64 8

64 8

Slide 10.1-7

Finding All Square Roots of a NumberCLASSROOM EXAMPLE 1

Find each square root.

Solution:

169

225

13

15

2564

2564

58

Slide 10.1-8

Finding Square RootsCLASSROOM EXAMPLE 2

Find the square of each radical expression.

Solution:

17 217 17

31 231 31

22 3x 222 3x 22 3x

Slide 10.1-9

Squaring Radical ExpressionsCLASSROOM EXAMPLE 3

Objective 2

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

Slide 10.1-10

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

25144

49

25 5

144 12

4 29 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 10.1-11

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 10.1-12

Identifying Types of Square RootsCLASSROOM EXAMPLE 4

Objective 3

Find cube, fourth, and other roots.

Slide 10.1-13

Find cube, fourth, and other roots.

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 10.1-14

Find each cube root.

3 64

3 27

3 512

4

3

8

Slide 10.1-15

Finding Cube Roots

Solution:

CLASSROOM EXAMPLE 5

Find each root.

4 81

4 81

4 81

5 243

5 243

3

3

Not a real number.

3

3

Solution:

Slide 10.1-16

Finding Other RootsCLASSROOM EXAMPLE 6

Objective 4

Graph functions defined by radical expressions.

Slide 10.1- 16

Square Root Function

The domain and range of the square root function are [0, ).

Slide 10.1- 17

Graph functions defined by radical expressions.

The domain and range of the cube function are (, ).

Slide 10.1- 18

Graph functions defined by radical expressions.

Cube Root Function

Graph the function by creating a table of values. Give the domain and range.

( ) 2f x x

x f(x)

–2

–1

0

2

2 2 0 1 2 1

0 2 1.41 2 2 2

Domain: [2, )

Range: [0, )Slide 10.1- 19

CLASSROOM EXAMPLE 7

Graphing Functions Defined with Radicals

Solution:

3( ) 1f x x

X f(x)

0

1

2

3

4

3 0 1 1 3 1 1 0

3 2 1 1 3 3 1 1.587

Domain: (, )

Range: (, )

3 4 1 1.44

Slide 10.1- 20

CLASSROOM EXAMPLE 7

Graphing Functions Defined with Radicals (cont’d)

Graph the function by creating a table of values. Give the domain and range.

Solution:

Objective 5

Find nth roots of nth powers.

Slide 10.1- 21

For any real number a,

That is, the principal square root of a2 is the absolute value of a.

2a2 | | .a a

Slide 10.1- 22

Find nth roots of nth powers.

Find each square root.

215 1|15 | 5 2( 12) | 12 12|

2y | |y 2( )y | | | |yy

Slide 10.1- 23

CLASSROOM EXAMPLE 8

Simplifying Square Roots by Using Absolute Value

Solution:

If n is an even positive integer, then

If n is an odd positive integer, then

That is, use absolute value when n is even; absolute value is not necessary when n is odd.

n na

| | .n na a

.n na a

Slide 10.1- 24

Find nth roots of nth powers.

Simplify each root.

44 ( 5) | | 55 55 ( 5) i5 s odd n

66 ( 3) | 3 | 3

84 m 2 is even m n

Slide 10.1- 25

CLASSROOM EXAMPLE 9

Simplifying Higher Roots by Using Absolute Value

Solution:

3 24x

186 y3| |y

8x3 66 ( )y

Objective 6

Use a calculator to find roots.

Slide 10.1- 26

Use a calculator to approximate each radical to three decimal places.

17 4.123 362 19.026

3 9482 21.166 4 6825 9.089

Slide 10.1- 27

CLASSROOM EXAMPLE 10

Finding Approximations for Roots

Solution:

In electronics, the resonant frequency f of a circuit may be found by the

formula where f is the cycles per second, L is in henrys, and C is

in farads. (Henrys and farads are units of measure in electronics). Find the

resonant frequency f if L = 6 10-5 and

C = 4 10-9.

12

fLC

12

fLC

5 9

1

2 (6 10 )(4 10 )f

324,874

About 325,000 cycles per second.

Slide 10.1- 28

CLASSROOM EXAMPLE 11

Using Roots to Calculate Resonant Frequency

Solution: