Beginning Algebra, 11th Edition

Using the Definition of

Evaluate.

(a) (b)

(c) - 32 4/5= - 132 1/5 24 = - 24 = - 16

9 3/2= 19 1/2 23 = 33 = 27 64 2/3= 164 1/3 22 = 42 = 16

EXAMPLE 2 am/n

SECTION 8.7 Using Rational Numbers as Exponents 541

If ais a nonnegative number and mand nare integers with then

am/n 1 a1/n 2 mA

n

2 aBm.

n 7 0,

am/n

If ais a positive number and mand nare integers with then

am/n

1

am/n

.

n 7 0,

am/n

The base is 32,

not 32.-

Be careful with

signs here. NOW TRY

Earlier, was defined as for nonzero numbers aand integers n. This

same result applies to negative rational exponents.

a-n a-n=a^1 n

Using the Definition of

Evaluate.

(a) (b) 27 - 4/3=

1

27 4/3

=

1

127 1/3 24

=

1

34

=

1

81

32 - 3/5=

1

32 3/5

=

1

132 1/5 23

=

1

23

=

1

8

EXAMPLE 3 a-m/n

Think:

32 1/5=^532 =2.

This is notthe

same as - 27 4/3. NOW TRY

OBJECTIVE 3 Apply the rules for exponents using rational exponents.

All the rules for exponents given earlier still hold when the exponents are fractions.

Using the Rules for Exponents with Fractional Exponents

Simplify. Write each answer in exponential form with only positive exponents.

EXAMPLE 4

CAUTION In Example 3(b),do notwrite as This is incorrect.

The negative exponent does not indicate a negative number.Also, the negative ex-

ponent indicates to use the reciprocal of the base,not the reciprocal of the exponent.

27 - 4/3 - 27 3/4.

NOW TRY
EXERCISE 2
Evaluate.

(a) (b)

(c) - 16 7/4

27 4/3 16 3/2

Keep the same base. Keep the same base.

(a) (b)

5 1/4

5 3/4

= 5 1/4-3/4= 5 - 2/4= 5 - 1/2=

1

5 1/2

3 2/3# 3 5/3= 3 2/3+5/3= 3 7/3

NOW TRY
EXERCISE 3
Evaluate.

(a) 16 - 3/4 (b) 8 - 2/3

NOW TRY ANSWERS

2. (a) 81 (b) 64 (c)


3. (a)^18 (b)^14
   
   
   • 128
   
   
   
   

This example suggests the following definition for am/n.