Rationalizing the Denominator
Example 1
Simplify:
Solution 

Factor the radicand. 

The radical in the denominator is a cube root. To rationalize the
denominator, we want to make the radicand a perfect cube.
 2 occurs twice as a factor. To make a perfect cube, we multiply by 2.
 w occurs once as a factor. To make a perfect cube, we multiply by w^{2}.
We multiply by 1 written in the form


Multiply the numerators and multiply
the denominators.


Simplify the denominator. 

So, 

Suppose we have a radical expression such as
Recall that when we multiply conjugates, the result is always a rational
number.
Therefore, to eliminate the radical in the denominator of the expression,
we multiply the numerator and denominator by
, the conjugate
of the denominator.
Example 2
Simplify:
Solution 

To rationalize the denominator
we multiply the numerator
and denominator by
, the conjugate of x


Multiply the numerators and
multiply the denominators.


Remove the parentheses. 

Simplify the denominator.
The two middle terms add
to zero. 

So, 

