Beginning Algebra, 11th Edition

540 CHAPTER 8 Roots and Radicals

OBJECTIVES OBJECTIVE 1 Define and use expressions of the form We now con-

sider how an expression such as should be defined so that all the rules for expo-

nents developed earlier still hold. Consider the following.

This agrees with the product rule for exponents from Section 5.1.By definition,

Since both and equal 5,

Similarly,

and

so.

These examples suggest the following definition.

5 1/3 should equal 235

235 # 235 # 235 = 2353 =5,

5 1/3# 5 1/3# 5 1/3= 5 1/3+1/3+1/3= 5 3/3=5,

5 1/2 should equal 25.

5 1/2# 5 1/2 25 # 25

A 25 BA 25 B=5.

5 1/2# 5 1/2= 5 1/2+1/2= 51 = 5

5 1/2

a1/n.

Using Rational Numbers as Exponents

8.7

1 Define and use

expressions of the

form

2 Define and use

expressions of the

form

3 Apply the rules for

exponents using

rational exponents.

4 Use rational

exponents to

simplify radicals.

am/n.

a1/n.

If ais a nonnegative number and nis a positive integer, then

a1/n.

n

2 a

a1/n

Notice that the denominator of the rational exponent is the index of the radical.

NOW TRY
EXERCISE 1
Simplify.

(a) (b)

(c) 256 1/4

144 1/2 729 1/3

NOW TRY ANSWERS

1. (a) 12 (b) 9 (c) 4

Using the Definition of

Simplify by first writing in radical form.

(a) By the definition of ,

(b) 27 1/3= 2327 = 3 (c) 216 1/3= 23216 = 6 (d) 64 1/6= 2664 = 2

16 1/2 a1/n 16 1/2= 216 =4.

EXAMPLE 1 a1/n

The denominator

is the index.

NOW TRY

OBJECTIVE 2 Define and use expressions of the form A more general

exponential expression, such as , can be defined using the power rule,

.

However, can also be written as follows. Same answer

Either way, the answer is the same. Taking the root first involves smaller numbers and

is often easier.

16 3/4= 11632 1/4= 140962 1/4= 244096 = 8

16 3/4

16 3/4= 116 1/4 23 = A (^2416) B^3 = 23 = 8
1 am 2 n=amn

16 3/4

am/n.

http://www.ebook777.com

http://www.ebook777.com