Beginning Algebra, 11th Edition

(c)

Square the binomial.

= 25 - 102 x+x NOW TRY

= 52 - (^2152) A 2 xB+ A 2 xB^2
A 5 - 2 xB
2
SECTION 8.5 More Simplifying and Operations with Radicals 525
(b)

= 28 + 1623

= 12 + 1623 + 16 A 223 B^2 = 4 # 3 = 12

= A 223 B

2

+ 2 A 223 B 142 + 42

A^223 +^4 B

2

Do nottry

to combine

further here.

CAUTION Only like radicals can be combined.InExamples 2(a) and (b),

59 - 14210 Z 45210 and 28 + 1623 Z 4423.

Example 3uses the rule for the product of the sum and difference of two terms.

Using a Special Product with Radicals

Find each product. Assume that

(a)

(b)

NOW TRY

In Example 3,the pairs of expressions and and

and are called conjugatesof each other.

OBJECTIVE 2 Use conjugates to rationalize denominators of radical

expressions.To rationalize the denominator in a quotient such as

we multiply the numerator and denominator by the conjugate of the denominator,

here , to obtain

The denominator contains no radicals. It has been rationalized.

2 A 4 + 23 B

A^4 -^23 BA^4 +^23 B

, or

2 A 4 + 23 B

13

.

4 + 23

2

4 - 23

,

2 x+ 26

4 + 23 4 - 23 2 x- 26

A^2 xB

(^2) =x;
=x- (^6) A (^26) B^2 = 6
= A 2 xB
2

• A 26 B
  2

A^2 x-^26 BA^2 x+^26 B

= 13

= 16 - 3 42 =16; A (^23) B^2 = 3

= 42 - A 23 B

2

A^4 +^23 BA^4 -^23 B

xÚ0.

EXAMPLE 3

1 xy 21 xy 2 x^2 y^2

NOW TRY
EXERCISE 2
Find each product. Assume
that

(a)

(b) A 3 + 2 yB
2

A 27 - 4 B^2

yÚ0.

NOW TRY
EXERCISE 3
Find each product. Assume that
.

(a)

(b) A 2 x+ 223 BA 2 x- 223 B

A 8 + 210 BA 8 - 210 B

xÚ 0

NOW TRY ANSWERS

2. (a)
   (b)


3. (a) 54
   (b)x- 12

9 + 62 y+y

23 - 827

Let .x= 223 and y= 4

1 x+y 22 =x^2 + 2 xy+y^2

Let and x= 4 y= 23.

1 x+y 21 x-y 2 =x^2 - y^2