Beginning Algebra, 11th Edition

OBJECTIVE 5 Simplify other roots. The product and quotient rules for radicals

also apply to other roots.

SECTION 8.2 Multiplying, Dividing, and Simplifying Radicals 509

(b)

Factor; 16 is a perfect fourth power.

Product rule

= 2242 Take the fourth root.

= 2416 # 242

= 2416 # 2

2432

(c)

Quotient rule

Take cube roots. NOW TRY

Other roots of radicals involving variables can also be simplified. To simplify

cube roots with variables, use the fact that for any real number a,

This is true whether ais positive, negative, or 0.

Simplifying Cube Roots Involving Variables

Simplify each radical.

(a)

(b)

Product rule

= 3 x^433 =27; 1 x^423 =x^12

= 2327 # 23 x^12

2327 x^12

=m^21 m^223 =m^6

23 m^6

EXAMPLE 9

23 a^3 a.

=

3

5

=

2327

23125

B

3

27

125

Remember to write

the root index 4 in

each radical.

Properties of Radicals

For all real numbers for which the indicated roots exist,

and

n

2 a

n

2 b

 n

B

a

b

1 b 02.

n

2 a#

n

2 b

n

2 ab

NOW TRY
EXERCISE 8
Simplify each radical.

(a) (b)

(c)
B

3

1

125

23250 2448

Simplifying Other Roots

Simplify each radical.

(a)

Factor; 8 is a perfect cube.

Product rule

= 2234 Take the cube root.

= 238 # 234

= 238 # 4

2332

EXAMPLE 8

NOW TRY ANSWERS

8. (a) 5232 (b) 2243 (c) 51

Remember to write

the root index 3 in

each radical.