Intermediate Algebra (11th edition)

We obtain an alternative definition of by using the power rule for exponents

differently than in the earlier definition. If all indicated roots are real numbers,

then am/n= am^1 1/n^2 = 1 am 2 1/n, so am/n= 1 am 2 1/n.

am/n

438 CHAPTER 8 Roots, Radicals, and Root Functions

If all indicated roots are real numbers, then

am/n 1 a1/n 2 m 1 am 2 1/n.

am/n

We can now evaluate an expression such as in two ways.

or

The result is the same.

In most cases, it is easier to use 1 a1/n 2 m.

27 2/3= 12722 1/3 = 729 1/3= 9

27 2/3 = 127 1/3 22 = 32 = 9

27 2/3

Radical Form of

If all indicated roots are real numbers, then

That is, raise ato the mth power and then take the nth root, or take the nth root of

aand then raise to the mth power.

am/n

n

2 amA

n

2 aB

m

.

am/n

For example,

and

so 8 2/3= 2382 = A 238 B

2

.

8 2/3= A 238 B

2

8 2/3= 2382 = 2364 = 4 , = 22 = 4 ,

OBJECTIVE 3 Convert between radicals and rational exponents. Using

the definition of rational exponents, we can simplify many problems involving radi-

cals by converting the radicals to numbers with rational exponents. After simplifying,

we can convert the answer back to radical form if required.

Converting between Rational Exponents and Radicals

Write each exponential as a radical. Assume that all variables represent positive real

numbers. Use the definition that takes the root first.

(a) (b) (c)

(d)

(e)

(f ) 1 a^2 +b^22 1/2= 2 a^2 + b^2

r-2/3=

1

r2/3

=

1

A^23 r^ B

2

6 x2/3- 14 x 2 3/5 = (^6) A 23 x (^) B
2

- A 254 xB

3

9 m5/8= (^9) A 28 m (^) B
5
6 3/4= A (^246) B
3

13 1/2= 213

EXAMPLE 4

2 a^2 +b^2 Za+b