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Copyright © 2010 Pearson Education, Inc. All rights reserved Sec 9.2 - 1
Rational Exponents
Copyright © 2010 Pearson Education, Inc. All rights reserved.
Rational Exponents
Exponents of the Form a1/n
a1/n
If is a real number, thenan
a1/n = .an
Copyright © 2010 Pearson Education, Inc. All rights reserved.
Evaluate each expression.
(a)
EXAMPLE 1 Evaluating Exponentials of the Form a1/n
Rational Exponents
271/3
(b) 641/2
=
=
(c) –6251/4
(d) (–625)1/4
Copyright © 2010 Pearson Education, Inc. All rights reserved.
Rational Exponents
Caution on Roots
(c) –6251/4
(d) (–625)1/4 =
EXAMPLE 1
=
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(e) (–243)1/5
Evaluate each expression.
EXAMPLE 1 Evaluating Exponentials of the Form a1/n
Rational Exponents
(f)1/2 4
25
Copyright © 2010 Pearson Education, Inc. All rights reserved.
Rational Exponents
Exponents of the Form am/n
am/n
If m and n are positive integers with m/n in lowest terms, then
am/n = ( a1/n ) m,
provided that a1/n is a real number. If a1/n is not a real number, then am/n
is not a real number.
Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 9.2 - 7
Evaluate each exponential.
(a) 253/2
EXAMPLE 2
(b) 322/5
(c) –274/3
(d) (–64)2/3
(e) (–16)3/2
Evaluate each exponential.
(a) 32–4/5
EXAMPLE 3 Evaluating Exponentials with Negative
Rational Exponents
Rational Exponents
Copyright © 2010 Pearson Education, Inc. All rights reserved.
Evaluate each exponential.
(b)
EXAMPLE 3 Evaluating Exponentials with Negative
Rational Exponents
Rational Exponents
–4/3 827 =
We could also use the rule = here, as follows.–mb
ama
b
Copyright © 2010 Pearson Education, Inc. All rights reserved.
Rational Exponents
Caution on Roots
CAUTION
When using the rule in Example 3 (b), we take the reciprocal only of the
base, not the exponent. Also, be careful to distinguish between exponential
expressions like –321/5, 32–1/5, and –32–1/5.
–321/5 = –2,1232–1/5 = ,
12and –32–1/5 = – .
Copyright © 2010 Pearson Education, Inc. All rights reserved.
Rational Exponents
Alternative Definition of am/n
am/n
If all indicated roots are real numbers, then
am/n = ( a1/n ) m = ( a m ) 1/n.
Copyright © 2010 Pearson Education, Inc. All rights reserved.
Rational Exponents
Radical Form of am/n
Radical Form of am/n
If all indicated roots are real numbers, then
In words, raise a to the mth power and then take the nth root, or take the
nth root of a and then raise to the mth power.
am/n = = ( ) .n
am n a m
Copyright © 2010 Pearson Education, Inc. All rights reserved.
Write each exponential as a radical. Assume that all variables represent
positive real numbers. Use the definition that takes the root first.
(a) 151/2
EXAMPLE 4 Converting between Rational Exponents
and Radicals
Rational Exponents
15= (b) 105/6 = ( )5106
(c) 4n2/3 = 4( )2n3
(d) 7h3/4 – (2h)2/5 = 7( )3h4
(e) g–4/5 =1
g4/5=
1
( )4g5
– ( )25 2h
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In (f) – (h), write each radical as an exponential. Simplify. Assume that all
variables represent positive real numbers.
EXAMPLE 4 Converting between Rational Exponents
and Radicals
Rational Exponents
(f) 33 = 331/2
= 76/3(g) 763 = 72 = 49
= m, since m is positive.(h) m55
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Rational Exponents
Rules for Rational Exponents
Rules for Rational Exponents
Let r and s be rational numbers. For all real numbers a and b for which the
indicated expressions exist:
ar · as = ar + s
( ar ) s = ar s ( ab ) r = ar br
a–r = 1ar = ar – sar
as =a
b
–rbr
ar
=ab
rar
br a–r = 1a
r .
Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 9.2 - 16
Write with only positive exponents. Assume that all variables represent
positive real numbers.
(a) 63/4 · 61/2
EXAMPLE 5 Applying Rules for Rational Exponents
9.2 Rational Exponents
(b) 32/3
35/6
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Write with only positive exponents. Assume that all variables represent
positive real numbers.
EXAMPLE 5
(c) m1/4 n–6
m–8 n2/3
–3/4
Copyright © 2010 Pearson Education, Inc. All rights reserved.
Write with only positive exponents. Assume that all variables represent
positive real numbers.
EXAMPLE 5
Rational Exponents
(d) x3/5(x–1/2 – x3/4)
Do not make the common mistake of multiplying exponents in the
first step.
Copyright © 2010 Pearson Education, Inc. All rights reserved.
Rational Exponents
Caution on Converting Expressions to Radical Form
CAUTION
Use the rules of exponents in problems like those in Example 5. Do not
convert the expressions to radical form.
Copyright © 2010 Pearson Education, Inc. All rights reserved.
Rewrite all radicals as exponentials, and then apply the rules for rational
exponents. Leave answers in exponential form. Assume that all variables
represent positive real numbers.
EXAMPLE 6 Applying Rules for Rational Exponents
Rational Exponents
(a) ·4a3 3
a2
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Rewrite all radicals as exponentials, and then apply the rules for rational
exponents. Leave answers in exponential form. Assume that all variables
represent positive real numbers.
EXAMPLE 6 Applying Rules for Rational Exponents
Rational Exponents
(b)4 c
c3
