Intermediate Algebra (11th edition)

We justify the product rule by using the rules for rational exponents. Since

and

2

n

a# 2

n

b=a1/n#b1/n= 1 ab 2 1/n= 2

n

ab.

2

n

2 b= b1/n,

n

a=a1/n

444 CHAPTER 8 Roots, Radicals, and Root Functions

CAUTION Use the product rule only when the radicals have the same index.

Quotient Rule for Radicals

If and are real numbers, and nis a natural number, then

That is, the nth root of a quotient is the quotient of the nth roots.

n

B

a

b



n

2 a

n

2 b

.

2 bZ 0,

n

2 b

n

a

Using the Quotient Rule

Simplify. Assume that all variables represent positive real numbers.

(a) (b)

(c) -ba=- ab

B

3 -

8

125

=

B

3

- 8

125

=

23 - 8

23125

=

- 2

5

=-

2

5

B

7

36

=

27

236

=

27

B 6

16

25

=

216

225

=

4

5

EXAMPLE 3

NOW TRY
EXERCISE 1
Multiply. Assume that all
variables represent positive
real numbers.

(a)

(b) 22 mn# 215

27 # 211

NOW TRY
EXERCISE 2
Multiply. Assume that all
variables represent positive
real numbers.

(a)

(b)

(c)

(d) 235 # 249

2720 x# 273 xy^3

245 t# 246 r^3

234 # 235

NOW TRY ANSWERS

1. (a) (b)


2. (a) (b)

(c)

(d)This expression cannot be

simplified by the product rule.

2760 x^2 y^3

2320 2430 tr^3

277 230 mn

Using the Product Rule

Multiply. Assume that all variables represent positive real numbers.

(a) (b) (c)

= 235 NOW TRY

= 25 # 7 = 211 p = 277 xyz

25 # 27 211 # 2 p 27 # 211 xyz

EXAMPLE 1

Using the Product Rule

Multiply. Assume that all variables represent positive real numbers.

(a) (b) (c)

(d) cannot be simplified using the product rule for radicals, because the

indexes and are different. NOW TRY

OBJECTIVE 2 Use the quotient rule for radicals.The quotient rule for rad-

icalsis similar to the product rule.

14 52

242 # 252

= 2336

= 233 # 12 = 2424 yr^2 = 2650 m^5

233 # 2312 248 y# 243 r^22610 m^4 # 265 m

EXAMPLE 2

Remember to

write the index.