8-6
8-6
Radical
Radical Expressions
Expressions and
and Rational
Rational Exponents
Exponents
Warm Up
Lesson Presentation
Lesson Quiz
HoltMcDougal
ALgebra2Algebra 2
Holt
8-6
Radical Expressions and Rational Exponents
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
8-6
Radical Expressions and Rational Exponents
Objectives
Rewrite radical expressions by using
rational exponents.
Simplify and evaluate radical expressions
and expressions containing rational
exponents.
Holt McDougal Algebra 2
8-6
Radical Expressions and Rational Exponents
Vocabulary
index
rational exponent
Holt McDougal Algebra 2
8-6
Radical Expressions and Rational Exponents
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 McDougal Algebra 2
8-6
Radical Expressions and Rational Exponents
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.
Holt McDougal Algebra 2
n
8-6
Radical Expressions and Rational Exponents
Reading Math
When a radical sign shows no index, it represents
a square root.
Holt McDougal Algebra 2
8-6
Radical Expressions and Rational Exponents
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.
Holt McDougal Algebra 2
8-6
Radical Expressions and Rational Exponents
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.
Holt McDougal Algebra 2
8-6
Radical Expressions and Rational Exponents
The properties of square roots in Lesson 1-3 also
apply to nth roots.
Holt McDougal Algebra 2
8-6
Radical Expressions and Rational Exponents
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.
Holt McDougal Algebra 2
8-6
Radical Expressions and Rational Exponents
Example 2A: Simplifying Radical Expressions
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.
8-6
Radical Expressions and Rational Exponents
Example 2B: Simplifying Radical Expressions
Quotient Property.
Simplify the numerator.
Rationalize the numerator.
Product Property.
Simplify.
Holt McDougal Algebra 2
8-6
Radical Expressions and Rational Exponents
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.
8-6
Radical Expressions and Rational Exponents
Check It Out! Example 2b
Simplify the expression. Assume that all
variables are positive.
x8
4
3
4
Quotient Property.
Rationalize the numerator.
Product Property.
4
27 2
x
3
Holt McDougal Algebra 2
Simplify.
8-6
Radical Expressions and Rational Exponents
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.
8-6
Radical Expressions and Rational Exponents
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.
Holt McDougal Algebra 2
8-6
Radical Expressions and Rational Exponents
Writing Math
The denominator of a rational exponent becomes
the index of the radical.
Holt McDougal Algebra 2
8-6
Radical Expressions and Rational Exponents
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 )
Write with a radical.
(–2)3
Evaluate the root.
–8
Evaluate the power.
5
Method 2 Evaluate
the power first.
3
Holt McDougal Algebra 2
Write with a radical.
5
-32,768 Evaluate the power.
–8
Evaluate the root.
8-6
Radical Expressions and Rational Exponents
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
Write with a radical.
Evaluate the root.
Evaluate the power.
Holt McDougal Algebra 2
3
(64 )1
Write will a radical.
3
64
Evaluate the power.
4
Evaluate the root.
8-6
Radical Expressions and Rational Exponents
Check It Out! Example 3b
Write the expression 4
simplify.
5
2
Method 1 Evaluate
the root first.
( 4)
Write with a radical.
(2)5
Evaluate the root.
2
32
5
Evaluate the power.
Holt McDougal Algebra 2
in radical form, and
Method 2 Evaluate
the power first.
2
2
(4 )5
Write with a radical.
1024
Evaluate the power.
32
Evaluate the root.
8-6
Radical Expressions and Rational Exponents
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
in radical form, and
Method 2 Evaluate
the power first.
Write with a radical.
(5)3
3
4
4
4
(625 )3
Write with a radical.
244,140,625 Evaluate the power.
125
Evaluate the root.
8-6
Radical Expressions and Rational Exponents
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
Holt McDougal Algebra 2
n
a
m
=a
m
n
Simplify.
8-6
Radical Expressions and Rational Exponents
Check It Out! Example 4
Write each expression by using rational
exponents.
a.
81
b.
3
4
c.
10
9
3
103
1000
Holt McDougal Algebra 2
5
Simplify.
5
2
4
1
2
Simplify.
8-6
Radical Expressions and Rational Exponents
Rational exponents have the same properties as
integer exponents (See Lesson 1-5)
Holt McDougal Algebra 2
8-6
Radical Expressions and Rational Exponents
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.
Holt McDougal Algebra 2
8-6
Radical Expressions and Rational Exponents
Example 5B: Simplifying Expressions with Rational
Exponents
Simplify each expression.
Quotient of Powers.
Simplify.
Negative Exponent Property.
1
4
Holt McDougal Algebra 2
Evaluate the power.
8-6
Radical Expressions and Rational Exponents
Example 5B Continued
Check Enter the expression in a graphing calculator.
Holt McDougal Algebra 2
8-6
Radical Expressions and Rational Exponents
Check It Out! Example 5a
Simplify each expression.
Product of Powers.
Simplify.
6
Evaluate the Power.
Check Enter the expression
in a graphing calculator.
Holt McDougal Algebra 2
8-6
Radical Expressions and Rational Exponents
Check It Out! Example 5b
Simplify each expression.
(–8)
1
–8
–
1
3
1
3
1
–
2
Negative Exponent Property.
Evaluate the Power.
Check Enter the
expression in a graphing
calculator.
Holt McDougal Algebra 2
8-6
Radical Expressions and Rational Exponents
Check It Out! Example 5c
Simplify each expression.
Quotient of Powers.
52
Simplify.
25
Evaluate the power.
Check Enter the expression
in a graphing calculator.
Holt McDougal Algebra 2
8-6
Radical Expressions and Rational Exponents
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 McDougal Algebra 2
8-6
Radical Expressions and Rational Exponents
Example 6 Continued
–
500 (200
t
1600
–
800
1600
–
1
2
) = 500(2
= 500(2
= 500(
1
2
)
)
1
2
Simplify.
)
500
=
1
2 2
≈ 354
The amount of radium-226 left
after 800 years would be about
354 mg.
Holt McDougal Algebra 2
Substitute 800 for t.
Negative Exponent
Property.
Simplify.
Use a calculator.
8-6
Radical Expressions and Rational Exponents
Check It Out! Music Application
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)?
= 64(2–1)
Use 64 cm for the length of the
string, and substitute 12 for n.
Simplify.
Negative Exponent Property.
Simplify.
= 32
The fret should be placed 32 cm from the bridge.
Holt McDougal Algebra 2
8-6
Radical Expressions and Rational Exponents
Lesson Quiz: Part I
Find all real roots.
1. fourth roots of 625
–5, 5
2. fifth roots of –243
–3
Simplify each expression.
3.
4.
2
5. Write (–216)
4
4y2
256y 8
2
3
in radical form and simplify.
2
3
-216 = 36
5
3
5
6. Write 21 using rational exponents. 21 3
Holt McDougal Algebra 2
(
)
8-6
Radical Expressions and Rational Exponents
Lesson Quiz: Part II
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?
$2300.01
Holt McDougal Algebra 2