Rationalizing the Denominator
Example 1
Simplify:
![](./articles_imgs/939/ration29.gif)
Solution |
![](./articles_imgs/939/ration30.gif) |
Factor the radicand. |
![](./articles_imgs/939/ration31.gif) |
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 w2.
We multiply by 1 written in the form
![](./articles_imgs/939/ration32.gif) |
![](./articles_imgs/939/ration33.gif) |
Multiply the numerators and multiply
the denominators.
|
![](./articles_imgs/939/ration34.gif) |
Simplify the denominator. |
![](./articles_imgs/939/ration35.gif) |
So, |
![](./articles_imgs/939/ration36.gif) |
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:
![](./articles_imgs/939/ration39.gif)
Solution |
![](./articles_imgs/939/ration40.gif) |
To rationalize the denominator
we multiply the numerator
and denominator by
, the conjugate of x
|
![](./articles_imgs/939/ration42.gif) |
Multiply the numerators and
multiply the denominators.
|
![](./articles_imgs/939/ration43.gif) |
Remove the parentheses. |
![](./articles_imgs/939/ration44.gif) |
Simplify the denominator.
The two middle terms add
to zero. |
![](./articles_imgs/939/ration45.gif) |
So, |
![](./articles_imgs/939/ration46.gif) |
|