Intermediate Algebra (11th edition)

436 CHAPTER 8 Roots, Radicals, and Root Functions

Evaluating Exponentials of the Form

Evaluate each exponential.

(a) (b)

(c)

(d) is not a real number, because the radicand, is

negative and the index is even.

(e) (f ) a

1

8

b

1/3

=

B

3

1

8

=

1

2

1 - 322 1/5= 25 - 32 =- 2

1 - 2562 1/4= 24 - 256 - 256,

- 256 1/4 = - 24256 = - 4

64 1/3= 2364 = 4 100 1/2 = 2100 = 10

EXAMPLE 1 a1/n
NOW TRY
EXERCISE 1
Evaluate each exponential.

(a) (b)

(c) (d)

(e) (f ) a

1

16

b

1/4

1 - 1252 1/3

- 625 1/4 1 - 6252 1/4

81 1/2 125 1/3

The denominator

is the index.

The denominator is

the index.

2 means.^22

NOW TRY

CAUTION Notice the difference between Examples 1(c) and (d).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.

OBJECTIVE 2 Define and use expressions of the form We know that

We can define a number like , where the numerator of the exponent is

not 1. For past rules of exponents to be valid,

Since

Generalizing from this example, we define am/nas follows.

8 2/3= A 238 B

2

= 22 =4.

8 1/3= 238 ,

8 2/3= 81 1/3^22 = 18 1/3 22.

8 1/3= 238. 8 2/3

am/n.

If mand nare positive integers with in lowest terms, then

provided that is a real number. If is not a real number, then is not a

real number.

a1/n a1/n am/n

am/n 1 a1/n 2 m,

m/n

am/n

Evaluating Exponentials of the Form

Evaluate each exponential.

(a) (b)

(c)

Because the base here is 4, the negative sign is notaffected by the exponent.

- 4 5/2 = - 14 5/2 2 = - 14 1/2 25 = - 1225 = - 32

36 3/2= 136 1/2 23 = 63 = 216 125 2/3= 1125 1/3 22 = 52 = 25

EXAMPLE 2 am>n

Be careful.

The base is 4.

Think:

125 1/ 3= 23125 = 5

Think:

36 1/2= 236 = 6

NOW TRY ANSWERS

1. (a) 9 (b) 5 (c)
   (d)It is not a real number.
   (e) - 5 (f )^12
   
   
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