Download a –m/n - BarsellaMath31

Chapter 8
Section 2
8.2 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 © 2012, 2008, 2004 Pearson Education, Inc.
Objective 1
Use exponential notation for nth
roots.
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Slide 8.2- 3
Use exponential notation for nth roots.
a1/n
If
n
a
is a real number, then
a1/ n  n a .
Notice that the denominator of the rational exponent is the index
of the radical.
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Slide 8.2- 4
CLASSROOM
EXAMPLE 1
Evaluating Exponentials of the Form a1/n
Evaluate each exponential.
Solution:
321/ 5
 5 32
2
641/ 2
 2 64
 64
811/ 4
  4 81
 3
(81)1/ 4
 4 81
Is not a real number because the radicand,
–81, is negative and the index, 4, is even.
(64)1/ 3
 3 64
 4
1

27
1

3
1/ 3
 1 
 
 27 
3
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
8
Slide 8.2- 5
Objective 2
Define and use expressions of
the form am/n.
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Slide 8.2- 6
Define and use expressions of the form am/n.
am/n
If m and n are positive integers with m/n in lowest terms, then
a
m/n

 a
1/ 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 © 2012, 2008, 2004 Pearson Education, Inc.
Slide 8.2- 7
CLASSROOM
EXAMPLE 2
Evaluating Exponentials of the Form am/n
Evaluate each exponential.
Solution:
25
3/ 2
  25
27
2/3
  27
16




1/ 2 3

1/ 3 2
3/ 2
 64 
2/3
 36 
3/ 2
  16

1/ 2 3
  64  


1/ 3

27 
3
 125
2
3
9
3
 43
 64
2
  4 
2
 16 
  64 
3
is not a real number, since
is not a real number.
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
 53
25

2
3
 36 
1/ 2
,
or
2
 16
36,
Slide 8.2- 8
Define and use expressions of the form am/n.
a–m/n
If
am/n
is a real number, then
a
m / n

1
a
m/n
(a  0).
A negative exponent does not necessarily lead to a negative result.
Negative exponents lead to reciprocals, which may be positive.
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Slide 8.2- 9
CLASSROOM
EXAMPLE 3
Evaluating Exponentials with Negative Rational Exponents
Evaluate each exponential.
Solution:
1
 3/ 4
81
3/ 4
81
36
3/ 2
 64 
 
 25 

1
 3/ 2
36
3/ 2

1

81 
1/ 4 3
1
 81
4
1

 36 
1/ 2 3

1
36
Copyright © 2012, 2008, 2004 Pearson Education, Inc.

3
1

27
1
 3
6
1

216
3
3


25
 25 
5
    
   
8
 64 
 64 
3/ 2
3
1
 3
3
125

512
Slide 8.2- 10
Define and use expressions of the form am/n.
am/n
If all indicated roots are real numbers, then
a
m/n

 a
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
1/ n
  a 
m
m
1/ n
.
Slide 8.2- 11
Define and use expressions of the form am/n.
Radical Form of am/n
If all indicated roots are real numbers, then
a
m/n
 a
n
m

 a
n
m
.
That is, 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 © 2012, 2008, 2004 Pearson Education, Inc.
Slide 8.2- 12
Objective 3
Convert between radicals and
rational exponents.
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Slide 8.2- 13
CLASSROOM
EXAMPLE 4
Converting between Rational Exponents and Radicals
Write each exponential as a radical. Assume that all variables,
represent positive real numbers.
Solution:
19
3/ 4
5x
2/3
3/ 5
x
2
3
  2x
y
2
3
3/ 5
5
1
 5/ 7 
x
x 5/ 7
2
1
4
11
14x
 19   19
  11 
 14  x 
 5 x    2x 

1/ 2

2 1/ 3
3
3
5
1
 x
7
5
 3 x2  y 2
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Slide 8.2- 14
CLASSROOM
EXAMPLE 4
Converting between Rational Exponents and Radicals (cont’d)
Write each radical as an exponential.
Solution:
37
4
8
8
9
z
8
 371/ 2
 98/ 4  92  81
= z, since z is assumed to be positive.
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Slide 8.2- 15
Objective 4
Use the rules for exponents with
rational exponents.
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Slide 8.2- 16
Use the rules for exponents with 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 a  a
r
s
r s
a 
r
r
s
a
a
r
1
 r
a
 ab 
rs
a a
 b   br
 
r
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
ar
r s
a
s
b
a
b
 
r
r
 a r br
r
b
 r
a
a
r
1
 
a
r
Slide 8.2- 17
CLASSROOM
EXAMPLE 5
Applying Rules for Rational Exponents
Write with only positive exponents. Assume that all variables
represent positive real numbers.
Solution:
1/ 2
3
1/ 2 1/ 3
3
3
1/ 3
2/3
7
4/3
7
a b

 b
1/ 3 2 / 3
7



6
2 / 3 4 / 3
 a b
3/ 6  2 / 6
3
7
2 / 3

1/ 3 2/ 31 6
a
1/36  1/36
b
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
 35 / 6
1
 2/3
7
 a b
   a  b 
1/ 3 1/ 3 6
6 / 3 6 / 3
a b
1/ 3 6
2 2
a b
1/ 3 6
2
a
 2
b
Slide 8.2- 18
CLASSROOM
EXAMPLE 5
Applying Rules for Rational Exponents (cont’d)
Write with only positive exponents. Assume that all variables
represent positive real numbers.
Solution:
 ab 
 2 1/ 5 
a b 
3 4
1/ 2
 a
 a
r
2/5
r
3/ 5
r
r
 b
5 1/ 2
8/5
2/ 5  3/ 5

b
21/ 5 1/ 2
a
5/ 2 21/10
r
r
r

r
2/ 5
2/ 5 8/ 5
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
 a b

r
b 21/10
 5/ 2
a
b
3/ 5
5/ 5
2/ 5
r

5 21/ 5 1/ 2
3 ( 2) 4 1/ 5 1/ 2
r
10/ 5
8/ 5
 r  r2
Slide 8.2- 19
CLASSROOM
EXAMPLE 6
Applying Rules for Rational Exponents
Write 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.
Solution:
4
 x 3/ 4 1/ 5  x15/ 20 4 / 20  x19 / 20
3/ 4
1/ 5

x

x
x  x
3
x5
3
x
3 6
5
x5/ 2
5/ 2 1/ 3
15/ 6  2/ 6
13/ 6
 1/ 3  x
x
x
x
 x
3
x
1/ 6
 x
Copyright © 2012, 2008, 2004 Pearson Education, Inc.

1/ 6 1/ 3
x
1/ 61/ 3
x
1/18
Slide 8.2- 20