Rationalizing the Denominator
Examples with Solutions
Since it is a basic property of radicals that
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:
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:
There really is nothing else we can do to this final expression to simplify it further, so it must be the required final answer.
Write the result of the following division in simplified form:
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
The only possible simplification at this stage is to rationalize the denominator, which can be accomplished by multiplying the numerator and denominator by :
as the required simplest final result.
Write the result of the following division in simplest form:
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.
Now, we rationalize the denominator by multiplying the numerator and denominator by to get
as the required final answer in simplified form.
The formula for the area, A, of a circle of radius r is given by
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.
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
we can write
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:
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:
Thus, the final simplified formula for r is