Properties Of Radicals, Simplifying Radicals, Rationalizing Operations Worksheet - Section A-7 Page 6
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9A-62
Appendix A A BASIC ALGEBRA REVIEW
(C)
( x
3)( x
5)
x x
3 x
5 x
15
x
2 x
15
3
3
3
3
3
3
3
3
2
2
3
2
2
3
(D)
(
m
n
)(
m
n)
m
m
n
mn
n
3
3
2
2
m
mn
m
n
n
Multiply and simplify.
M A T C H E D P R O B L E M
5
(A)
3( 6
4)
(B)
( 3
2)( 3
4)
3
3
3
3
2
2
(C)
( y
2)( y
4)
(D)
(
x
y
)(
x
y)
Rationalizing Operations
We now turn to algebraic fractions involving radicals in the denominator. Elimi-
nating a radical from a denominator is referred to as rationalizing the denomi-
nator. To rationalize the denominator, we multiply the numerator and denomina-
tor by a suitable factor that will rationalize the denominator—that is, will leave
the denominator free of radicals. This factor is called a rationalizing factor. The
following special products are of use in finding some rationalizing factors (see
Example 6, parts C and D):
2
2
(a
b)(a
b)
a
b
(1)
2
2
3
3
(a
b)(a
ab
b
)
a
b
(2)
2
2
3
3
(a
b)(a
ab
b
)
a
b
(3)
E x p l o r e / D i s c u s s
Use special products in equations (1) to (3) to find a rationalizing factor
for each of the following:
2
3
3
3
3
(A)
a
b
(B)
a
b
(C)
a
b
(D)
a
b
E X A M P L E
Rationalizing Denominators
6
Rationalize denominators.
2
3
2a
x
y
1
3
(A)
(B)
(C)
(D)
3
2
5
3b
3 x
2 y
m
2
2
(A)
5
is a rationalizing factor for
5
, since
5 5
5
5
. Thus,
S o l u t i o n s
we multiply the numerator and denominator by
5
to rationalize the
denominator:
3
3
5
3 5
5
5
5
5
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