Simplifying a Fractional Expression
Containing Radicals
Simplify:
3
9

x
2
+
x
2
3
9

x
2
9

x
2
.
EXAMPLE 11
1
x
+
7

1
x
7
.
h.
= 
1
x
1
x
+
h
2
,
h
Z
0,
x
Z
0,
x
Z 
h
x

x

h
= 
h.
=

h
1
h
1
x
1
x
+
h
2
=
x

x

h
hx
1
x
+
h
2
1
x
#
x
1
x
+
h
2
=
x
+
h.
1
x
+
h
x
1
x
+
h
2
=
x
=
x

1
x
+
h
2
hx
1
x
+
h
2
=
1
x
+
h
#
x
1
x
+
h
2

1
x
#
x
1
x
+
h
2
h
x
1
x
+
h
2
x
1
x
+
h
2
, h
Z
0, x
Z
0, x
Z 
h.
=
a
1
x
+
h

1
x
b
x
1
x
+
h
2
h
x
1
x
+
h
2
1
x
+
h

1
x
h
x
1
x
+
h
2
.
h
1
x
+
h
,
1
x
,
1
x
+
h

1
x
h
.
EXAMPLE 10
Simplify fractional expressions
that occur in calculus.
78
Chapter P
Prerequisites: Fundamental Concepts of Algebra
Solution
The least common denominator
is
Multiply the numerator and the
denominator by
Use the distributive property in the
numerator.
In the denominator:
Because the original expression was
in radical form, write the denominator
in radical form.
Check Point
11
Simplify:
Another fractional expression that you will encounter in calculus is
Can you see that this expression is not defined if
However, in calculus, you
will ask the following question:
What happens to the expression as
takes on values that get closer and closer
to 0, such as
and so on?
The question is answered by first
rationalizing the numerator
.This process involves
rewriting the fractional expression as an equivalent expression in which the numer
ator no longer contains any radicals.
To rationalize a numerator, multiply by 1 to
eliminate the radicals in the
numerator
. Multiply the numerator and the denominator
by the conjugate of the numerator.
Rationalizing a Numerator
Rationalize the numerator:
Solution
The conjugate of the numerator is
If we multiply the
numerator and denominator by
the simplified numerator will not
contain a radical.Therefore, we multiply by 1, choosing
for 1.
1
x
+
h
+
1
x
1
x
+
h
+
1
x
1
x
+
h
+
1
x
,
1
x
+
h
+
1
x
.
2
x
+
h

1
x
h
.
EXAMPLE 12
h
=
0.1,
h
=
0.01,
h
=
0.001,
h
=
0.0001,
h
h
=
0?
2
x
+
h

1
x
h
.
1
x
+
1
1
x
x
.
=
9
3
1
9

x
2
2
3
=
1
9

x
2
2
3
2
.
1
9

x
2
2
1
1
9

x
2
2
1
2
=
1
9

x
2
2
1
+
1
2
=
1
9

x
2
2
+
x
2
1
9

x
2
2
3
2
=
3
9

x
2
3
9

x
2
+
x
2
3
9

x
2
3
9

x
2
1
9

x
2
2
3
9

x
2
3
9

x
2
.
=
3
9

x
2
+
x
2
3
9

x
2
9

x
2
#
3
9

x
2
3
9

x
2
3
9

x
2
.
3
9

x
2
+
x
2
3
9

x
2
9

x
2
Rationalize numerators.
Section P.6
Rational Expressions
79
Multiply by 1.
and
Simplify:
Divide both the numerator and
denominator by
What happens to
as
gets closer and closer to 0? In Example
12, we showed that
As
gets closer to 0, the expression on the right gets closer to
or
Thus, the fractional expression
approaches
as
gets closer to 0.
Check Point
12
Rationalize the numerator:
2
x
+
3

1
x
3
.
h
1
2
1
x
2
x
+
h

1
x
h
1
2
1
x
.
1
1
x
+
1
x
,
1
2
x
+
0
+
1
x
=
h
2
x
+
h

1
x
h
=
1
2
x
+
h
+
1
x
.
h
2
x
+
h

1
x
h
h
.
=
1
2
x
+
h
+
1
x
,
h
Z
0
x
+
h

x
=
h
.
=
h
h
A
2
x
+
h
+
1
x
B
1
1
x
2
2
=
x
.
A
2
x
+
h
B
2
=
x
+
h
=
x
+
h

x
h
A
2
x
+
h
+
1
x
B
A
1
a
B
2

A
2
b
B
2
A
1
a

2
b
B A
1
a
+
2
b
B
=
=
A
2
x
+
h
B
2

1
1
x
2
2
h
A
2
x
+
h
+
1
x
B
2
x
+
h

1
x
h
=
2
x
+
h

1
x
h
#
2
x
+
h
+
1
x
2
x
+
h
+
1
x
Exercise Set P.6
Practice Exercises
In Exercises 1–6, find all numbers that must be excluded from the
domain of each rational expression.