Rationalizing Binomial DenominatorsRationalize each denominator.
(a)
=- 3 A 1 - 22 B, or - 3 + 322 Distributive property
=
3
- 1
(^) A 1 - (^22) B
=
(^3) A 1 - (^22) B
- 1
=
3 A 1 - 22 B
A^1 +^22 BA^1 -^22 B
3
1 + 22
EXAMPLE 5
462 CHAPTER 8 Roots, Radicals, and Root Functions
NOW TRY
EXERCISE 5
Rationalize each denominator.
(a) (b)
(c)
(d)
3 xZy, x 7 0, y 708
23 x- 2 y,
23 + 27
25 - 22
4
5 + 27
4
1 + 23
The denominator
is now a rational
number.(b)
Multiply in the denominator.Subtract in the denominator.Notice that the numerator is left in factored form. This makes it easier to determine
whether the expression is written in lowest terms.
=
(^5) A 4 + (^23) B
13
=
5 A 4 + 23 B
16 - 3
=
(^5) A 4 + (^23) B
A^4 -^23 BA^4 +^23 B
5
4 - 23
Multiply the numerator and
denominator by the
conjugate of the denominator.= 1 - 2 , or - 1= 12 - A (^22) B^2
A^1 +^22 BA^1 -^22 B
1 - 22 ,
Again, we are multiplying
by a form of 1.
(c)
Multiply.= Subtract in the denominator.
210 - 26 - 215 + 3
2
=
210 - 26 - 215 + 3
5 - 3
=
A 2 - 3 BA 5 - 3 B
A 5 + 3 BA 5 - 3 B
22 - 23
25 + 23
Multiply the numerator and
denominator by 4 + 23.(d) ,
= Multiply in the denominator.
3 A 25 m+ 2 pB
5 m- p
=
(^3) A 25 m+ 2 pB
A 25 m- 2 pBA 25 m+ 2 pB
5 mZp, m 7 0, p 70
3
25 m- 2 p
Multiply the numerator and
denominator by 25 - 23.NOW TRY ANSWERS
- (a)
(b)(c)(d)(^8) A 23 x+ 2 y (^) B
3 x-y
215 + 26 + 235 + 214
3
(^2) A 5 - (^27) B
9
- 2 A 1 - 23 B, or - 2 + 223
Multiply the numerator and
denominator by 25 m+ 2 p.NOW TRY