A. Introduction
1. If we are performing a computation and obtain an answer of
, how are we likely to
write that answer? _____
Change the following fractions to simplest form:
2. We are used to the convention of writing a fraction is
in its simplest form. When fractions
contain square roots , they also are generally expected to be written in a
“standard” form.
The standard in this case is that a fraction may not contain any radicals in the
denominator . (The radicals that we will be working with are square roots.) When
radicals
do occur in the denominator (in fractions such as
or 
or
), we must carry out a
process known as rationalizing the denominator .
3. Equivalent Fractions
In order to understand the mathematics behind rationalizing the denominator, we
should
look to a familiar example of writing equivalent fractions .
If we were asked to write another fraction which is equivalent to
, we might, for
example, write 
. These
fractions are equivalent since they have the same effect when
used in calculations .
We can see below how we mathematically obtain
as an equivalent fraction to :
Notice that we multiplied by
,and
equals 1. We can multiply any
number by 1 without changing the value of the number.
4. Multiplying Square Roots
Before we learn how to rationalize a denominator , we must also remember how to
multiply square roots.
Example: Multiply the following and simplify your answers .
B. Rationalizing the Denominator
1. Consider the fraction
. To rationalize the
denominator, we multiply by
, as
follows:
Thus, our final answer is
.
Some may debate as to whether or not
looks any “ simpler ” than
; nonetheless, this
has become the standard. (Remember that your final answers ARE allowed to have
square roots in the numerator , but your answers may NOT contain square roots in
the
denominator.)
2. Consider the fraction 
. To rationalize
the denominator, we multiply by 
, as
follows:
Thus, our final answer is
.
3. Examples: Rationalize the denominators in the following fractions:
