Intermediate Algebra (11th edition)

(f )

Difference of squares

and NOW TRY

NOTEIn Example 1(d),we could have used the formula for the square of a binomial

to obtain the same result.

Apply the exponents. Multiply.

Add.

OBJECTIVE 2 Rationalize denominators with one radical term. As defined

earlier, a simplified radical expression has no radical in the denominator. The origin

of this agreement no doubt occurred before the days of high-speed calculation, when

computation was a tedious process performed by hand.

For example, consider the radical expression. To find a decimal approxima-

tion by hand, it is necessary to divide 1 by a decimal approximation for such as

1.414. It is much easier if the divisor is a whole number. This can be accomplished by

multiplying by 1 in the form. Multiplying by 1 in any form does not change

the value of the original expression.

Multiply by 1;

Now the computation requires dividing 1.414 by 2 to obtain 0.707, a much easier task.

With current technology, either form of this fraction can be approximated with

the same number of keystrokes. See FIGURE 10, which shows how a calculator gives

the same approximation for both forms of the expression.

(^22) = 1
22

1

22

#^22

22

=

22

2

22 22

1 22

22 ,

1 22

= 16 - 627

= 7 - 627 + 9

= A (^27) B 1 x-y 22 =x^2 - 2 xy+y^2
2

(^2) A (^27) B 132 + 32

A^27 -^3 B

2

=k-y, kÚ 0 yÚ 0

= A 2 kB

2

- A 2 yB

2

A 2 k+ 2 yBA 2 k- 2 yB

SECTION 8.5 Multiplying and Dividing Radical Expressions 459

FIGURE 10

Rationalizing the Denominator

A common way of “standardizing” the form of a radical expression is to have the

denominator contain no radicals. The process of removing radicals from a de-

nominator so that the denominator contains only rational numbers is called

rationalizing the denominator.This is done by multiplying by a form of 1.

Rationalizing Denominators with Square Roots

Rationalize each denominator.

(a)

Multiply the numerator and denominator by This is, in effect, multiplying

by 1.

=

327

7

3

27

=

3 # 27 27 # 27

27.

3

27

EXAMPLE 2

In the denominator,

The final denominator is now a rational number.

27 # 27 = 27 # 7 = 249 =7.

NOW TRY
EXERCISE 1
Multiply, using the FOIL
method.

(a)

(b)

(c)

(d)

(e)

mÚ0 and nÚ 0

A 2 m- 2 n BA 2 m+ 2 n B,

A 8 + 235 BA 8 - 235 B

A 215 - 4 B

2

A 27 + 25 BA 27 - 25 B

A 8 - 25 BA 9 - 22 B

NOW TRY ANSWERS

(a)
(b) 2 (c)
(d) 64 - 2325 (e)m-n

31 - 8215

72 - 822 - 925 + 210