Rationalizing the Denominator
Examples with Solutions
Example 1:
Simplify ![](./articles_imgs/945/pic22.gif)
solution:
Since it is a basic property of radicals that
![](./articles_imgs/945/pic23.gif)
we see that the given expression actually amounts to a
fraction with a radical in the denominator. Therefore, to
accomplish simplification of this expression, we need to
rationalize the denominator:
![](./articles_imgs/945/pic24.gif)
A quick check before moving on indicates that there is a
perfect square factor in the square root that remains, and so we
remove this factor from the square root before declaring a final
answer:
![](./articles_imgs/945/pic25.gif)
There really is nothing else we can do to this final
expression to simplify it further, so it must be the required
final answer.
Example 2:
Write the result of the following division in simplified form:
![](./articles_imgs/945/pic26.gif)
solution:
We deal with division involving radicals in detail in a later
document in this series. However, you are already familiar with
the way fractions express division, so that here we can write
![](./articles_imgs/945/pic27.gif)
The only possible simplification at this stage is to
rationalize the denominator, which can be accomplished by
multiplying the numerator and denominator by :
![](./articles_imgs/945/pic29.gif)
as the required simplest final result.
Example 3:
Write the result of the following division in simplest form:
![](./articles_imgs/945/pic30.gif)
solution:
This example is very similar to the previous Example 6, and so
you should use it as a practice problem. Attempt to work out the
final answer yourself before looking at the steps of our
solution, which follows.
Here
![](./articles_imgs/945/pic31.gif)
Now, we rationalize the denominator by multiplying the
numerator and denominator by to get
![](./articles_imgs/945/pic33.gif)
![](./articles_imgs/945/pic34.gif)
as the required final answer in simplified form.
Example 4:
The formula for the area, A, of a circle of radius r is given
by
![](./articles_imgs/945/pic35.gif)
Derive a formula for the radius, r, in terms of the area, A,
and make sure that your result satisfies the usual criteria of
simplicity.
solution:
We deal systematically with the topic of rearranging formulas
in another document on this website. However, we can handle this
problem without having to invoke a general strategy. Since
![](./articles_imgs/945/pic35.gif)
we can write
![](./articles_imgs/945/pic36.gif)
by just dividing both sides of the original formula by . Now we have a formula for r 2. To get a
formula for r, we need to take the square root of both sides:
![](./articles_imgs/945/pic37.gif)
This formula looks quite simple already, but it does consist
of the square root of a fraction, and so the usual rules of
simplification say we should rationalize the denominator. This is
easily done:
![](./articles_imgs/945/pic38.gif)
Thus, the final simplified formula for r is
![](./articles_imgs/945/pic39.gif)
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