A fraction that contains a radical in its
denominator can be written
as an equivalent fraction with a
rational denominator (a denominator
without a radical).

Never leave a radical in the
denominator of a fraction.
Always
rationalize the denominator. 
Situation 1  Monomial
Denominator 
When the denominator is a monomial
(one term),
multiply both the numerator and the
denominator by whatever makes the denominator an expression that can be
simplified so that it no longer contains a radical.
* Sometimes the value being multiplied happens to be exactly the
same as the denominator,
as in this first example (Example 1):
Example 1:
Simplify 

* Sometimes it is necessary to multiply by whatever makes the
denominator
a perfect square or
perfect cube or any other power that can be
simplified,
as seen in the next examples (Examples 2 and
3).
Example 2:
Simplify 

Example 3:
Simplify 



Multiplying by
will create the smallest perfect cube under the radical. Replacing
by
3, rationalizes the denominator. 

Make sure you
multiply by whatever makes the radicand
(the number under the
radical sign) the smallest possible value to be simplified. This will
avoid having to further simplify later on.

Situation 2  More than
One Term in Denominator 
When there is more than one term in the
denominator, the process is a little tricky. You will need to multiply
the numerator and denominator by the the
denominator's
conjugate. The
conjugate is the same expression as the
denominator but with the opposite sign in the middle,
separating the terms.
Example 4:
Simplify 


Multiply top and bottom by the conjugate of the denominator,
.
Notice that you are multiplying by 1, which does not change the
original expression. When multiplying the denominators in this problem, distribute
or use FOIL.
Notice what is happening to the middle terms when you
multiply the denominators. The middle terms will drop
out. Also, the last term has created a perfect square
under the square root.
If possible, always reduce your final answer. In this
problem, a factor of 2 can be removed from the top and bottom.
Did you notice that in this problem, we never distributed the
2 in the numerator? It is often best to work with the denominator
first, and then see what else needs to be done.

Example 5:
Simplify 


Multiply top and bottom by the conjugate of the denominator,
.
In this problem, we will need to multiply out both the numerator
and the denominator. Again you can use the distributive
method or FOIL.
Notice that while the middle terms are going to drop out on
the bottom, they are not going to drop out on the top.
This is OK. We just want the radical gone from the bottom.
Combine terms. The answer has a radical in the
numerator. This is OK. The bottom does NOT have a
radical, which was our goal. 

Be sure to enclose
expressions with multiple terms in parentheses.
This will help you to
remember to FOIL
(or distribute)
these expressions. 
Situation 3  Working with a
Reciprocal 
When working
with the reciprocal of an expression containing a radical,
it may be necessary to
rationalize the denominator.
Example 6:
Write the reciprocal of 


Here is our starting expression. The reciprocal
is created by inverting the numerator and denominator of the
starting expression.
Since we now have a radical in the denominator, we must
rationalize this denominator. Multiply top and bottom by the
conjugate of the denominator,
.
Multiply the denominators in this problem, by using the
distribute method or FOIL.
Again, we see the middle terms dropping out.
Simplify and combine.
Remember, that a radical in the top is OK. 
Radicals that are simplified
have: 


 no fractions left under the radical symbol.
 no perfect power factors in the
radicand, k.
 no exponents in the radicand, k, greater than
the index, n. 
 no radicals appearing in the denominator of a fractional answer.


