Warm Up
Simplify each expression.
16,807
121
729
1. 73 • 72
3. (32)3
2. 118
116
4.
5.
75
207
2 357
5 3
Rewrite radical expressions by using rational exponents.
Simplify and evaluate radical expressions and expressions containing rational exponents.
Objectives
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.
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
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.
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
Example 2B: Simplifying Radical Expressions
Quotient Property.
Product Property.
Simplify the numerator.
Rationalize the numerator.
Simplify.
Simplify the expression. Assume that all variables are positive.
Check It Out! Example 2a
Product Property.
Simplify.
Factor into perfect fourths.
2 • x2x
4 416x
4 24 •4 x4
4 24 •x4
Check It Out! Example 2b
Quotient Property.
Product Property.
Rationalize the numerator.
Simplify.
Simplify the expression. Assume that all variables are positive.
84
4 3
x
4227
3x
Check It Out! Example 2c
3 37 2x xg
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
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.
Write with a radical.
–8
Evaluate the power.
Evaluate the root.
( )−3
5 32
−5 32,768
Method 1 Evaluate the root first.
(4)1
Write with a radical.
6413
4
Evaluate the root.
Evaluate the power.
Check It Out! Example 3a
Write the expression in radical form, and simplify.
Method 2 Evaluate the power first.
Write will a radical.
4
Evaluate the power.
Evaluate the root.
( )13 64 ( )13 64
3 64
Method 1 Evaluatethe root first.
(2)5
Write with a radical.
452
32
Evaluate the root.
Evaluate the power.
Check It Out! Example 3b
Write the expression in radical form, and simplify.
Method 2 Evaluatethe power first.
Write with a radical.
32
Evaluate the power.
Evaluate the root.
( )52 4 ( )52 4
2 1024
Method 1 Evaluatethe root first.
(5)3
Write with a radical.
62534
125
Evaluate the root.
Evaluate the power.
Check It Out! Example 3c Write the expression in radical form, and simplify.
Method 2 Evaluate the power first.
Write with a radical.
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 a=m
mn na a
Simplify. 512
c.
24 5 =
mmn na a
Example 5A: Simplifying Expressions with Rational Exponents
Product of Powers.
Simplify each expression.
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 each expression.
Simplify.
Negative Exponent Property.
1 4
Evaluate the power.
Product of Powers.
Simplify each expression.
Simplify.
Evaluate the Power.6
Check It Out! Example 5a
Check Enter the expression in a graphing calculator.
Simplify each expression.
(–8)–13
Check It Out! Example 5b
1 –8
13
1 2
–
Negative Exponent Property.
Evaluate the Power.
Check Enter the expression in a graphing calculator.
Quotient of Powers.
Simplify each expression.
Simplify.
Evaluate the power.
52
Check It Out! Example 5c
25
Check Enter the expression in a graphing calculator.
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?
Example 6 ContinuedSubstitute 800 for t.
Simplify.
Negative Exponent Property.
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.
Negative Exponent Property.
Use 64 cm for the length of the string, and substitute 12 for n.
Check It Out! Music Application
= 64(2–1)
Simplify.= 32The 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.
Lesson Quiz: Part I
Find all real roots.
–5, 51. fourth roots of 625
2. fifth roots of –243 –3
Simplify each expression.
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
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
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