10.2 Rational
Exponents
Objective 1
Use exponential notation for nth roots.
Slide 10.2- 2
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.
Slide 10.2- 3
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
8
Slide 10.2- 4
Objective 2
Define and use expressions of the
form am/n.
Slide 10.2- 5
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.
Slide 10.2- 6
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 
 53
 125
2
3
9
3
 43
 64
2
  4 
25
3
2
 16 
  64 

2
3
3
is not a real number, since
is not a real number.
 36 
1/ 2
,
or
2
 16
36,
Slide 10.2- 7
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.
Slide 10.2- 8
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

3
1

27
1
 3
6
1

216
3
3


25
5
 25 
    
   
8
 64 
 64 
3/ 2
3
1
 3
3
125

512
Slide 10.2- 9
Define and use expressions of the form am/n.
am/n
If all indicated roots are real numbers, then
a
m/n

 a
1/ n
  a 
m
m
1/ n
.
Slide 10.2- 10
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.
Slide 10.2- 11
Objective 3
Convert between radicals and
rational exponents.
Slide 10.2- 12
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
Slide 10.2- 13
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|
Slide 10.2- 14
Objective 4
Use the rules for exponents with
rational exponents.
Slide 10.2- 15
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
a a
 b   br
 
r
s
a
a
r
ar
r s
a
s
a
1
 r
a
 ab 
rs
r
b
a
 b   ar
 
r
 a r br
r
a
r
1
 
a
r
Slide 10.2- 16
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
 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 10.2- 17
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
 a b

2/ 5
2/ 5 8/ 5
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 10.2- 18
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
x
5
x5/ 2
5/ 2 1/ 3
15/ 6  2 / 6
13/ 6
 1/ 3  x
x
x
x
 x
3
1/ 6
 x

1/ 6 1/ 3
x
1/ 61/ 3
x
1/18
Slide 10.2- 19