Holt McDougal Algebra 2 5-6 Radical Expressions and Rational Exponents 5-6 Radical Expressions and...

Holt McDougal Algebra 2

5-6 Radical Expressions and Rational Exponents 5-6 Radical Expressions and Rational Exponents

Holt ALgebra2

Warm Up

Lesson Presentation

Lesson Quiz



5-6 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



Rewrite radical expressions by using rational exponents.

Simplify and evaluate radical expressions and expressions containing rational exponents.

Objectives



indexrational exponent

Vocabulary



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



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

Reading Math



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.



Find all real roots.

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.



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



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!



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



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



Check It Out! Example 2c

3 37 2x x

Product Property of Roots.

Simplify.

Simplify the expression. Assume that all variables are positive.

x3

3 9x



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



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

Writing Math



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



Method 1 Evaluate the root first.

(4)1


6413

4

Evaluate the root.

Evaluate the power.

Check It Out! Example 3a

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



Method 1 Evaluatethe root first.

(2)5


452

32

Evaluate the root.

Evaluate the power.

Check It Out! Example 3b


Method 2 Evaluatethe power first.


32

Evaluate the power.

Evaluate the root.

( )52 4 ( )52 4

2 1024



Method 1 Evaluatethe root first.

(5)3


62534

125

Evaluate the root.

Evaluate the power.

Check It Out! Example 3c




125

Evaluate the power.

Evaluate the root.

( )34 625 ( )34 625

4 244,140,625



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



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



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



Example 5A: Simplifying Expressions with Rational Exponents

Product of Powers.


Simplify.

Evaluate the Power.

72

49

Check Enter the expression in a graphing calculator.



Example 5B: Simplifying Expressions with Rational Exponents

Quotient of Powers.


Simplify.

Negative Exponent Property.

1 4

Evaluate the power.

Holt McDougal Algebra 2 5-6 Radical Expressions and Rational Exponents 5-6 Radical Expressions and...

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