Beginning Algebra, 11th Edition

OBJECTIVE 3 Rationalize denominators with cube roots. To rationalize a

denominator with a cube root, we change the radicand in the denominator to a perfect

cube.

Rationalizing Denominators with Cube Roots

Rationalize each denominator.

(a)

First write the expression as a quotient of radicals. Then multiply numerator and

denominator by a sufficient number of factors of 2 to make the radicand in the de-

nominator a perfect cube. This will eliminate the radical in the denominator. Here,

multiply by

B

3

3

2

=

233

232

=

233 # 232 # 2

232 # 232 # 2

=

233 # 2 # 2

232 # 2 # 2

=

2312

2

232 # 2 , or 2322.

B

3

3

2

EXAMPLE 5

520 CHAPTER 8 Roots and Radicals

CAUTION A common error in a problem like the one in Example 5(a)is to

multiply by instead of Doing this would give a denominator of

Because 4 is not a perfect cube, the denominator is still not rationalized.

232 # 232 = 234.

232 2322.

Denominator radicand is a perfect

cube.

232 # 2 # 2 = 2323 = 2

(b)

Since multiply numerator and denominator by a sufficient num-

ber of factors of 2 to get a perfect cube in the radicand in the denominator.

(c)

Multiply numerator and denominator by a sufficient number of factors of 3 and

of xto get a perfect cube in the radicand in the denominator.

232

233 x^2

=

232 # 233 # 3 #x

233 #x#x# 233 # 3 #x

=

2318 x

2313 x 23

=

2318 x

3 x

232

233 x^2

, xZ 0

233

234

=

233 # 232

232 # 2 # 232

=

236

232 # 2 # 2

=

236

2

234 = 232 # 2 ,

233

234

We need 3 factors

of 2 in the radicand

in the denominator.

NOW TRY

Denominator radicand is a perfect

cube.

= 3 x

233 #x#x# 233 # 3 #x= 2313 x 23

We need 3 factors

of 3 and 3 factors

of xin the radicand

in the denominator.

NOW TRY
EXERCISE 5
Rationalize each denominator.

(a) (b)

(c)

234

239 t

, tZ 0

232

B 235

3

2

7

NOW TRY ANSWERS

5. (a) (b)

(c)

2312 t^2

3 t

2350

5

2398

7

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