#### Transcript Holt McDougal Algebra 2 5-6

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Expressions and
and Rational
Rational Exponents
Exponents
Warm Up
Lesson Presentation
Lesson Quiz
HoltMcDougal
ALgebra2Algebra 2
Holt
5-6
Warm Up
Simplify each expression.
1. 73 • 72
16,807
118
116
121
3. (32)3
729
4. 75
5 3
20
2 35
7
2.
5.
7
Holt McDougal Algebra 2
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Objectives
rational exponents.
and expressions containing rational
exponents.
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Vocabulary
index
rational exponent
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The nth root of a real number a can be written as
, where n is the index
When a number has more than one root, the radical
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When a radical sign shows no index, it represents
a square root.
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Example 1: Finding Real Roots
Find all 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.
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Check It Out! Example 1
Find all real roots.
a. fourth roots of –256
A negative number has no real fourth roots.
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.
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Remember!
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.
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Simplify each expression. Assume that all
variables are positive.
Factor into perfect fourths.
Product Property.
3xxx
3x3
Holt McDougal Algebra 2
Simplify.
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Check It Out! Example 2a
Simplify the expression. Assume that all
variables are positive.
4
4
16x 4
4
24 • x4
4
24 •x4
2x
2x
Holt McDougal Algebra 2
Factor into perfect fourths.
Product Property.
Simplify.
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Check It Out! Example 2c
Simplify the expression. Assume that all
variables are positive.
3
x7
3
3
x2
x9
x3
Holt McDougal Algebra 2
Product Property of Roots.
Simplify.
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A rational exponent is an exponent that can be
expressed as m , where m and n are integers and
n
n ≠ 0. Radical expressions can be written by using
rational exponents.
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Writing Math
The denominator of a rational exponent becomes
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Example 3: Writing Expressions in Radical Form
3
5
Write the expression (–32) in radical form
and simplify.
Method 1 Evaluate
the root first.
( -32 )
(–2)3
Evaluate the root.
–8
Evaluate the power.
5
Method 2 Evaluate
the power first.
3
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-32,768 Evaluate the power.
–8
Evaluate the root.
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Check It Out! Example 3a
1
3
Write the expression 64 in radical form, and
simplify.
Method 1 Evaluate
the root first.
( 64 )
Method 2 Evaluate
the power first.
1
3
(4)1
4
Evaluate the root.
Evaluate the power.
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(64 )1
3
64
Evaluate the power.
4
Evaluate the root.
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Check It Out! Example 3b
Write the expression 4
simplify.
5
2
Method 1 Evaluate
the root first.
( 4)
(2)5
Evaluate the root.
2
32
5
Evaluate the power.
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Method 2 Evaluate
the power first.
2
2
(4 )5
1024
Evaluate the power.
32
Evaluate the root.
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Check It Out! Example 3c
Write the expression 625
simplify.
Method 1 Evaluate
the root first.
(
4
)
3
625
Evaluate the root.
125
Evaluate the power.
Holt McDougal Algebra 2
Method 2 Evaluate
the power first.
(5)3
3
4
4
4
(625 )3
244,140,625 Evaluate the power.
125
Evaluate the root.
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Example 4: Writing Expressions by Using Rational
Exponents
Write each expression by using rational
exponents.
A.
B.
13
13
4
8
1
2
n
a
m
=a
Simplify.
m
n
3
15
5
33
27
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n
a
m
=a
m
n
Simplify.
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Check It Out! Example 4
Write each expression by using rational
exponents.
a.
81
b.
3
4
c.
10
9
3
103
1000
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Simplify.
5
2
4
1
2
Simplify.
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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
Simplify each expression.
Product of Powers.
72
Simplify.
49
Evaluate the Power.
Check Enter the
expression in a
graphing calculator.
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