Rationalizing the Denominator
A. What It Means to Rationalize the Denominator
In order that all of us doing math can compare answers, we
agree upon a common conversation, or set of rules, concerning the
form of the answers.
For instance, we could easily agree that we would not leave an
answer in the form of 3 + 4, but would write 7 instead.
When the topic switches to that of radicals, those doing math
have agreed that a RADICAL IN SIMPLE FORM will not (among other
things) have a radical in the denominator of a fraction. We will
all change the form so there is no radical in the denominator.
Now a radical in the denominator will not be something as
simple as . Instead, it will have a radicand which
will not come out from under the radical sign like .
Since is an irrational number, and we need to
make it NOT irrational, the process of changing its form so it is
no longer irrational is called RATIONALIZING THE DENOMINATOR.
B. There are 3 Cases of Rationalizing the Denominator
1. Case I : There is ONE TERM in the
denominator and it is a SQUARE ROOT.
2. Case II : There is ONE TERM in the
denominator, however, THE INDEX IS GREATER THAN TWO. It might be
a cube root or a fourth root.
3. Case III : There are TWO TERMS in the
denominator.
Let's study Case III:
3. Case III : There are TWO TERMS in the
denominator.
Example :
Procedure : We will multiply both top and
bottom by the conjugate. The conjugate is the same two terms but
with a different sign between them.
Since the conjugate for this numerator is , we will multiply top and bottom by that number.
Note how our use of the conjugate on the bottom is recreating
the special product of The Product of Conjugate Binomials, as in:
(x + 3)(x  3) Which will always result in squaring everything to
get x^{ 2} 9. And, squaring everything would get rid of
the term with the square root.
You might need to review the product of two conjugate
binomials if you can't see how we got the denominator.
Note: We could distribute the square root of
three in the numerator, but there doesn't seem to be any
advantage in doing so. Also, we don't have any factors we can
cancel, so this is the answer.
Review of the Product of the Conjugate Binomials
(x + 3)(x  3) 
is the product of the conjugate binomials
because we see the same two terms with different signs
between them. This could also be called "The
product of the sum of two terms and the difference of the
two terms."
This, then, would be the sum of x and 3 times the
difference of x and 3.

The product will always be:
The Square of the First Term Minus the Square of the
Second Term.
In this example, x^{ 2}  9 will be our answer.
