Beginning Algebra, 11th Edition

SECTION 8.4 Rationalizing the Denominator 517

OBJECTIVES

Rationalizing the Denominator

8.4

1 Rationalize

denominators with

square roots.

2 Write radicals in

simplified form.

3 Rationalize

denominators with

cube roots.

OBJECTIVE 1 Rationalize denominators with square roots.Although cal-

culators now make it fairly easy to divide by a radical in an expression such as , it

is sometimes easier to work with radical expressions if the denominators do not con-

tain any radicals.

For example, the radical in the denominator of can be eliminated by multiply-

ing the numerator and denominator by since

Multiply by

This process of changing the denominator from a radical (an irrational number)

to a rational number is called rationalizing the denominator.The value of the rad-

ical expression is not changed. Only the form is changed, because the expression

has been multiplied by 1 in the form.

Rationalizing Denominators

Rationalize each denominator.

(a)

Multiply by

Lowest terms

(b)

12

28

=

326

2

=

926

6

(^26) =1.
26

=

9 # 26

26 # 26

9

26

EXAMPLE 1

22

22

(^22) =1.
22

1

22

=

1 # 22

22 # 22

=

22

2

22 , 22 # 22 = 24 =2.

1

22

1

22

NOW TRY
EXERCISE 1
Rationalize each denominator.

(a) (b)

3

224

15

25

NOW TRY ANSWERS

1. (a) (b)
   26

(^3254)
In the denominator,
26 # 26 = 236 = 6.
The denominator could be rationalized by multiplying by
However, simplifying the denominator first is more direct.

28.

Multiply by

Multiply.

= 322 124 =3;lowest terms NOW TRY

=

1222

4

= 22 # 22 = 24 = 2

12 # 22

2 # 2

(^22) =1.
22

=

12 # 22

222 # 22

= 28 = 24 # 22 = 222

12

222

12

28