Rational Exponents

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Sharp Tutor Mon, Jan 17, 2022 07:12 PM UTC

OBJECTIVES:
1. Use exponential notation for nth roots.
2. Define and use expressions of the form a^m/n.
3. Convert between radicals and rational exponents.
4. Use the rules for exponents with rational exponents.

1. Copyright © 2010 Pearson Education, Inc. All rights reserved
Sec 9.2 - 1

2. Chapter 9
Roots, Radicals, and Root
Functions
Copyright © 2010 Pearson Education, Inc. All rights reserved
Sec 9.2 - 2

3. 9.2
Rational Exponents
Copyright © 2010 Pearson Education, Inc. All rights reserved
Sec 9.2 - 3

4. 9.2 Rational Exponents
Objectives
1. Use exponential notation for nth roots.
2. Define and use expressions of the form am/n.
3. Convert between radicals and rational exponents.
4. Use the rules for exponents with rational exponents.
Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 9.2 - 4

5. 9.2 Rational Exponents
Exponents of the Form a1/n
n
If a is a real number, then
n
a1/n = a.
Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 9.2 - 5

6. 9.2 Rational Exponents
EXAMPLE 1 Evaluating Exponentials of the Form a1/n
Evaluate each expression.
3
(a) 271/3 = 27 = 3
(b) 641/2 = 64 = 8
4
(c) –625 1/4 = – 625 = –5
4
(d) (–625)1/4 = –625 is not a real number because the radicand,
–625, is negative and the index is even.
Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 9.2 - 6

7. 9.2 Rational Exponents
Caution on Roots
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.
EXAMPLE 1
4
(c) –625 1/4 = – 625 = –5
4
(d) (–625) 1/4 = –625 is not a real number because the radicand,
–625, is negative and the index is even.
Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 9.2 - 7

8. 9.2 Rational Exponents
EXAMPLE 1 Evaluating Exponentials of the Form a1/n
Evaluate each expression.
5
(e) (–243)1/5 = –243 = –3
4 1/2 4 2
(f) = =
25 25 5
Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 9.2 - 8

9. 9.2 Rational Exponents
Exponents of the Form 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 - 9

10. 9.2 Rational Exponents
EXAMPLE 2 Evaluating Exponentials of the Form am/n
Evaluate each exponential.
(a) 253/2 = ( 251/2 )3 = 53 = 125
(b) 322/5 = ( 321/5 )2 = 22 = 4
(c) –274/3 = –( 27)4/3 = –( 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.
Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 9.2 - 10

11. 9.2 Rational Exponents
EXAMPLE 3 Evaluating Exponentials with Negative
Rational Exponents
Evaluate each exponential.
(a) 32–4/5
By the definition of a negative exponent,
–4/5
1
32 = 4/5
.
32
5 4
Since 32 4/5
= 32 = 24 = 16,
1 1
= 32 –4/5
= =
324/5 16 .
Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 9.2 - 11

12. 9.2 Rational Exponents
EXAMPLE 3 Evaluating Exponentials with Negative
Rational Exponents
Evaluate each exponential.
8 –4/3 1 1 1 1 81
(b) = = = = =
27 8 4/3 8 4 2 4 16 16
3 81
27 27 3
b –m a m
We could also use the rule = here, as follows.
a b
27 4 3 4 81
8 –4/3 27 4/3
= = 3 = =
27 8 8 2 16
Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 9.2 - 12

13. 9.2 Rational Exponents
Caution on Roots
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.
1 1
–321/5 = –2, 32–1/5 = , and –32–1/5 = – .
2 2
Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 9.2 - 13

14. 9.2 Rational Exponents
Alternative Definition of 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. Sec 9.2 - 14

15. 9.2 Rational Exponents
Radical Form of am/n
Radical Form of am/n
If all indicated roots are real numbers, then
n n
am/n = = (am ) . a m
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.
Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 9.2 - 15

16. 9.2 Rational Exponents
EXAMPLE 4 Converting between Rational Exponents
and Radicals
Write each exponential as a radical. Assume that all variables represent
positive real numbers. Use the definition that takes the root first.
= 15 6 10 5
(a) 151/2 (b) 105/6 = ( )
3 n 2
(c) 4n2/3 = 4( )
4 h 3 5 2h 2
(d) 7h3/4 – (2h)2/5 = 7( ) – ( )
1 1
(e) g –4/5 = =
g4/5 5 g 4
( )
Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 9.2 - 16

17. 9.2 Rational Exponents
EXAMPLE 4 Converting between Rational Exponents
and Radicals
In (f) – (h), write each radical as an exponential. Simplify. Assume that all
variables represent positive real numbers.
(f) 33 = 331/2
3
(g) 76 = 76/3 = 72 = 49
(h) 5 = m, since m is positive.
m5
Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 9.2 - 17

18. 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:
1 ar a –r r
r
a · a = a s r+s
a –r
= r s = a
r–s = br
a a b a
r 1
a = ar
r r
(a ) r s
= a rs
( ab ) r
= a b r r
a –r
= .
b b a
Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 9.2 - 18

19. 9.2 Rational Exponents
EXAMPLE 5 Applying Rules for Rational Exponents
Write with only positive exponents. Assume that all variables represent
positive real numbers.
(a) 63/4 · 61/2 = 63/4 + 1/2 = 65/4 Product rule
(b) 32/3 1
= 32/3 – 5/6 = 3–1/6 = Quotient rule
35/6 31/6
Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 9.2 - 19

20. 9.2 Rational Exponents
EXAMPLE 5 Applying Rules for Rational Exponents
Write with only positive exponents. Assume that all variables represent
positive real numbers.
(c) m1/4 n–6 –3/4
= (m1/4)–3/4 (n–6)–3/4
m–8 n2/3 ( m–8)–3/4 (n2/3)–3/4
= m–3/16 n9/2 Power rule
m6 n–1/2
= m–3/16 – 6 n9/2 – (–1/2) Quotient rule
= m–99/16 n5
n5 Definition of negative
=
m99/16 exponent
Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 9.2 - 20

21. 9.2 Rational Exponents
EXAMPLE 5 Applying Rules for Rational Exponents
Write with only positive exponents. Assume that all variables represent
positive real numbers.
(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.
Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 9.2 - 21

22. 9.2 Rational Exponents
Caution on Converting Expressions to Radical Form
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. Sec 9.2 - 22

23. 9.2 Rational Exponents
EXAMPLE 6 Applying Rules for Rational Exponents
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.
4 3 3/4 2/3
(a) a 3 · a2 = a · a Convert to rational exponents.
= a3/4 + 2/3 Product rule
= a9/12 + 8/12 Write exponents with a common
denominator
= a17/12
Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 9.2 - 23

24. 9.2 Rational Exponents
EXAMPLE 6 Applying Rules for Rational Exponents
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.
4
(b) c = c1/4 Convert to rational exponents.
c3/2
c3
= c1/4 – 3/2 Quotient rule
= c1/4 – 6/4 Write exponents with a common
denominator
= c–5/4
1
= Definition of negative exponent
5/4
c
Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 9.2 - 24

25. 9.2 Rational Exponents
EXAMPLE 6 Applying Rules for Rational Exponents
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.
(c) 5 3 = 5
x2 x2/3
= ( x2/3 )1/5
= x2/15
Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 9.2 - 25

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