holt algebra 2 8-6 radical expressions and rational exponents 8-6 radical expressions and rational...

Holt Algebra 2

8-6 Radical Expressions and Rational Exponents 8-6 Radical Expressions and Rational Exponents

Holt ALgebra2

Warm Up

Lesson Presentation

Lesson Quiz

Holt Algebra 2

8-6 Radical Expressions and Rational Exponents

Warm Up

Simplify each expression.

16,807

121

729

1. 73 • 72

3. (32)3

2. 118

116

4.

5.

75

20

7

2 357

5 3

Holt Algebra 2


Rewrite radical expressions by using rational exponents.

Simplify and evaluate radical expressions and expressions containing rational exponents.

Objectives

Holt Algebra 2


indexrational exponent

Vocabulary

Holt Algebra 2


You are probably familiar with finding the square root of a number. These two operations are inverses of each other. Similarly, there are roots that correspond to larger powers.

5 and –5 are square roots of 25 because 52 = 25 and (–5)2 = 25

2 is the cube root of 8 because 23 = 8.

2 and –2 are fourth roots of 16 because 24 = 16 and (–2)4 = 16.

a is the nth root of b if an = b.

Holt Algebra 2


The nth root of a real number a can be written as the radical expression , where n is the index (plural: indices) of the radical and a is the radicand. When a number has more than one root, the radical sign indicates only the principal, or positive, root.

n a

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When a radical sign shows no index, it represents a square root.

Reading Math

Holt Algebra 2


Find all real roots.

Example 1: Finding Real Roots

A. sixth roots of 64

A positive number has two real sixth roots. Because 26 = 64 and (–2)6 = 64, the roots are 2 and –2.

B. cube roots of –216

A negative number has one real cube root. Because (–6)3 = –216, the root is –6.

C. fourth roots of –1024

A negative number has no real fourth roots.

Holt Algebra 2



a. fourth roots of –256

A negative number has no real fourth roots.

Check It Out! Example 1

b. sixth roots of 1

A positive number has two real sixth roots. Because 16 = 1 and (–1)6 = 1, the roots are 1 and –1.

c. cube roots of 125

A positive number has one real cube root. Because (5)3 = 125, the root is 5.

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The properties of square roots in Lesson 1-3 also apply to nth roots.

Holt Algebra 2


When an expression contains a radical in the denominator, you must rationalize the denominator. To do so, rewrite the expression so that the denominator contains no radicals.

Remember!

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Simplify each expression. Assume that all variables are positive.

Example 2A: Simplifying Radical Expressions

Factor into perfect fourths.

Product Property.

Simplify.

3 x x x

3x3

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Example 2B: Simplifying Radical Expressions

Quotient Property.

Product Property.

Simplify the numerator.

Rationalize the numerator.

Simplify.

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Simplify the expression. Assume that all variables are positive.

Check It Out! Example 2a

Product Property.

Simplify.

Factor into perfect fourths.

2 x

2x

4 416x

4 24 •4 x4

4 24 •x4

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Check It Out! Example 2b

Quotient Property.

Product Property.

Rationalize the numerator.

Simplify.


84

4 3

x

4227

3x

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Check It Out! Example 2c

3 37 2x x

Product Property of Roots.

Simplify.


x3

3 9x

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A rational exponent is an exponent that can be expressed as , where m and n are integers and n ≠ 0. Radical expressions can be written by using rational exponents.

mn

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The denominator of a rational exponent becomes the index of the radical.

Writing Math

Holt Algebra 2


Example 3: Writing Expressions in Radical Form

Method 1 Evaluate the root first.

(–2)3

Write with a radical.

Write the expression (–32) in radical form and simplify.

35

–8

Evaluate the root.

Evaluate the power.

Method 2 Evaluate the power first.


–8

Evaluate the power.

Evaluate the root.

( )-3

5 32

-5 32,768

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Method 1 Evaluate the root first.

(4)1


6413

4

Evaluate the root.

Evaluate the power.


Write the expression in radical form, and simplify.


Write will a radical.

4

Evaluate the power.

Evaluate the root.

( )13 64 ( )13 64

3 64

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Method 1 Evaluatethe root first.

(2)5


452

32

Evaluate the root.

Evaluate the power.



Method 2 Evaluatethe power first.


32

Evaluate the power.

Evaluate the root.

( )52 4 ( )52 4

2 1024

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Method 1 Evaluatethe root first.

(5)3


62534

125

Evaluate the root.

Evaluate the power.





125

Evaluate the power.

Evaluate the root.

( )34 625 ( )34 625

4 244,140,625

Holt Algebra 2


Example 4: Writing Expressions by Using Rational Exponents

Write each expression by using rational exponents.

Simplify.

15 5 3

1312 Simplify.

A. B.

4813

33

27

=m

mn na a =m

mn na a

Holt Algebra 2


Write each expression by using rational exponents.

Simplify.

a. b.

3481

103

Check It Out! Example 4

9310

1000

m

mn na am

mn na a

Simplify. 512

c.

24 5

mmn na a

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Rational exponents have the same properties as integer exponents (See Lesson 1-5)

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Example 5A: Simplifying Expressions with Rational Exponents

Product of Powers.


Simplify.

Evaluate the Power.

72

49

Check Enter the expression in a graphing calculator.

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Example 5B: Simplifying Expressions with Rational Exponents

Quotient of Powers.


Simplify.

Negative Exponent Property.

1 4

Evaluate the power.

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Example 5B Continued


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Product of Powers.


Simplify.

Evaluate the Power.6



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(–8)–13


1 –8

13

1 2

–


Evaluate the Power.


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Quotient of Powers.


Simplify.

Evaluate the power.

52


25


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Example 6: Chemistry Application

Radium-226 is a form of radioactive element

that decays over time. An initial sample of

radium-226 has a mass of 500 mg. The mass of

radium-226 remaining from the initial sample

after t years is given by . To the

nearest milligram, how much radium-226 would

be left after 800 years?

Holt Algebra 2


Example 6 Continued

Substitute 800 for t.

Simplify.


800 1600500 (200– ) = 500(2– )

t1600

= 500( ) 1

2 12

12 = 500(2– )

= 500

2 12

Simplify.

Use a calculator.≈ 354The amount of radium-226 left after 800 years would be about 354 mg.

Holt Algebra 2



Use 64 cm for the length of the string, and substitute 12 for n.

Check It Out! Music Application

= 64(2–1)

Simplify.= 32

The fret should be placed 32 cm from the bridge.

To find the distance a fret should be place from the bridge

on a guitar, multiply the length of the string by ,

where n is the number of notes higher that the string’s

root note. Where should the fret be placed to produce the

E note that is one octave higher on the E string (12 notes

higher)?

Simplify.

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Lesson Quiz: Part I


–5, 51. fourth roots of 625

2. fifth roots of –243 –3


24. 4y2

5. Write (–216) in radical form and simplify.23

6. Write using rational exponents. 5321

3. 84 256y

= 36( )-2

3 2163 521

Holt Algebra 2


7. If $2000 is invested at 4% interest compounded

monthly, the value of the investment after t years

is given by . What is the value of the

investment after 3.5 years?

Lesson Quiz: Part II

$2300.01

holt algebra 2 8-6 radical expressions and rational exponents 8-6 radical expressions and rational...

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