copyright © 2010 pearson education, inc. all rights reserved sec 9.2 - 1 rational exponents
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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
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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
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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
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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
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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
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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 = – .
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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.
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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
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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
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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.
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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.
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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
(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