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Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Sec 9.2 - 1
9.2 Rational Exponents
Objectives
1. Use exponential notation for nth roots.
2. Define am/n.
3. Convert between radicals and rational exponents.4. Use the rules for exponents with rational exponents.
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Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Sec 9.2 - 2
9.2 Rational Exponents
Exponents of the Form a1/ n
a1/ n
If is a real number, then a n
a1/ n
= . a
n
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Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Sec 9.2 - 3
Evaluate each expression.
(a)
EXAMPLE 1 Evaluating Exponentials of the Form a1 /n
9.2 Rational Exponents
271/3 27 3
= 3
(b) 641/2 64 = 8
=
=
(c) –6251/4 625 4= –5= –
(d) ( –625)1/4 –6254
is not a real number because the radicand,
–625, is negative and the index is even.
=
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Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Sec 9.2 - 4
9.2 Rational Exponents
Caution on Roots
CAUTION
Notice the difference between parts (c) and (d) in Example 1. The radical
in part (c) is the negative fourth root of a positive number, while the radical
in part (d) is the principal fourth root of a negative number, which is not a real number.
(c) –6251/4 625 4 = –5= –
(d) ( –625)1/4 –6254
is not a real number because the radicand,
–625, is negative and the index is even.
=
EXAMPLE 1
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Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Sec 9.2 - 5
(e) ( –243)1/5 –2435
= –3=
Evaluate each expression.
EXAMPLE 1 Evaluating Exponentials of the Form a1 /n
9.2 Rational Exponents
(f) 1/2 4
25=
425
=25
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Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Sec 9.2 - 6
9.2 Rational Exponents
Exponents of the Form a m/ n
a m/ n
If m and n are positive integers with m/n in lowest terms, then
a m/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.
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Evaluate each exponential.
(a) 253/2
EXAMPLE 2 Evaluating Exponentials of the Form a m/n
9.2 Rational Exponents
= ( 251/2 )3 = 53 = 125
(b) 322/5 = ( 321/5 )2 = 22 = 4
(c) –274/3 = –( 271/3 )4 = –(3)4 = –81
(d) ( –
64)2/3 = [( –
64)1/3 ]2 = ( –
4)2 = 16
(e) ( –16)3/2 is not a real number, since ( –16)1/2 is not a real number.
= –( 27)4/3
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Evaluate each exponential.
(a) 32 –4/5
EXAMPLE 3 Evaluating Exponentials with Negative
Rational Exponents
9.2 Rational Exponents
1
324/5
By the definition of a negative exponent,
32 –4/5 = .
Since 324/5 =4
= 24 = 16,5 32
= 32 –4/5 = .1
324/5=
116
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Evaluate each exponential.
(b)
EXAMPLE 3 Evaluating Exponentials with Negative
Rational Exponents
9.2 Rational Exponents
–4/3 827 4/3 8
27
1= =
827
34
1
23
= 4
11681
= 1 81
16=
We could also use the rule = here, as follows. –m b
a
m a
b
–4/3 827
4/3 278
=278
3= 4
= 4 3
28116
=
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9.2 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, 12
32 –1/5 = , 12
and –32 –1/5 = – .
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9.2 Rational Exponents
Alternative Definition of a m/ n
a m/ n
If all indicated roots are real numbers, then
a m/n
= ( a1 / n
) m
= ( a m
)1 / n
.
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9.2 Rational Exponents
Radical Form of a m/ n
Radical Form of a m/ n
If all indicated roots are real numbers, then
In words, we can raise to the power and then take the root, or take the
root and then raise to the power.
a m/n = = ( ) . n a m 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
9.2 Rational Exponents
15= (b) 105/6 = ( )510 6
(c) 4n2/3 = 4( )2n3
(d) 7h3/4
–
(2h)2/5
= 7( )3
h4
(e) g –4/5 = 1
g4/5=
1
( )4g5
–( )
252h
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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
9.2 Rational Exponents
(f) 33 = 331/2
= 76/3(g) 763 = 72 = 49
= m, since m is positive. (h) m55
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Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Sec 9.2 - 15
9.2 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:
a r · a s = a r + s
( a r
) s
= a r s
( ab ) r
= a r
b
r
a – r =
1 a
r = a r – s a
r a
s = a b
– r b r
a r
= a
b
r a r
b r a
– r
=
1
a
r
.
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Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 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
= 63/4 + 1/2 = 65/4 Product rule
Quotient rule = 32/3 – 5/6(b) 32/3
35/6= 3 –1/6 1
31/6=
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Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Sec 9.2 - 17
Write with only positive exponents. Assume that all variables represent
positive real numbers.
EXAMPLE 5 Applying Rules for Rational Exponents
9.2 Rational Exponents
Power rule
(c) m1/4 n –6
m –8 n2/3
–3/4=
( m –8) –3/4 (n2/3) –3/4(m1/4) –3/4 (n –6) –3/4
=
m6 n –1/2 m –3/16 n9/2
= m –3/16 – 6 n9/2 – ( –1/2) Quotient rule
= m –99/16 n5
Definition of negative
exponent =
m99/16 n5
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Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Sec 9.2 - 18
Write with only positive exponents. Assume that all variables represent
positive real numbers.
EXAMPLE 5 Applying Rules for Rational Exponents
9.2 Rational Exponents
(d) x3/5( x –1/2 – x3/4) = x3/5 · x –1/2
– x3/5 · x3/4 Distributive property
= x3/5 + ( –1/2) – x3/5 + 3/4 Product rule
= x1/10 – x27/20
Do not make the common mistake of multiplying exponents in the
first step.
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Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Sec 9.2 - 19
9.2 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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Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Sec 9.2 - 20
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
9.2 Rational Exponents
Convert to rational exponents. (a) · 4 a3 3 a2 = a3/4 · a2/3
= a3/4 + 2/3
= a
9/12 + 8/12
= a17/12
Product rule
Write exponents with a commondenominator
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Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Sec 9.2 - 21
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
9.2 Rational Exponents
Convert to rational exponents.
Quotient rule
Write exponents with a common
denominator
(b)
4
c
c3= c
1/4
c3/2
= c1/4 – 3/2
= c1/4 – 6/4
= c –5/4
=
c5/4
1
Definition of negative exponent
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Copyright © 2006 Pearson Education Inc Publishing as Pearson Addison-Wesley Sec 9.2 - 22
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
9.2 Rational Exponents
(c) 3 x25 x2/35=
( x2/3 )1/5=
x2/15=
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