Beginning Algebra, 11th Edition

Using the Product Rule to Multiply Radicals

Use the product rule for radicals to find each product.

EXAMPLE 1

SECTION 8.2 Multiplying, Dividing, and Simplifying Radicals 505

(a)

= 26

= 22 # 3

22 # 23 (b)

= 235

27 # 25 (c)

Assume .aÚ 0

= 211 a

211 # 2 a (d)

= 31

231 # 231

OBJECTIVE 2 Simplify radicals by using the product rule.A square root

radical is simplified when no perfect square factor other than 1 remains under

the radical symbol.

Using the Product Rule to Simplify Radicals

Simplify each radical.

(a)

Factor; 4 is a perfect square.

Product rule in the form

= 225 24 = 2

= 24 # 25 2 a#b= 2 a# 2 b

= 24 # 5

220

EXAMPLE 2

NOW TRY

20 has a perfect

square factor of 4.

Thus, Because 5 has no perfect square factor (other than 1),

is called the simplified formof Note that represents a product whose

factors are 2 and

We could also factor 20 into prime factors and look for pairs of like factors. Each

pair of like factors produces one factor outside the radical in the simplified form.

(b)

Factor; 36 is a perfect square.

Product rule

= 622 236 = 6

= 236 # 22

= 236 # 2

272

220 = 22 # 2 # 5 = 225

25.

220. 225

220 = 225. 225

We could also factor 72 into its prime factors and look for pairs of like factors.

In either case, we obtain as the simplified form of

(c)

Factor; 100 is a perfect square.

Product rule

(d) The number 15 has no perfect square factors (except 1), so

cannot be simplified further.

215 215

= 1023 2100 = 10

= 2100 # 23

= 2100 # 3

2300

622 272.

272 = 22 # 2 # 2 # 3 # 3 = 2 # 3 # 22 = 622

Look for the greatest

perfect square factor of 72.

NOW TRY
EXERCISE 1
Find each product.

(a)

(b)

(c) 27 # 2 k, kÚ 0

211 # 211

25 # 211

NOW TRY
EXERCISE 2
Simplify each radical.

(a) (b)

(c) 285

228 299

NOW TRY ANSWERS

1. (a) (b) 11 (c)


2. (a) (b)
   (c)It cannot be simplified
   further.

227 3211

255 27 k

NOW TRY