It is considered bad practice to have a radical in the denominator of a fraction. When this happens we multiply the numerator and denominator by the same thing in order to clear the radical. In the lesson on dividing radicals we talked about how this was done with monomials. Here we will look at how this is done with binomials.
If the binomial is in the numerator the process to rationalize the denominator is essentially the same as with monomials The only difference is we will have to distribute in the numerator.
If the binomial occurs in the denominator we will have to use a different strategy to clear the radical. Consider , if we were to multiply the denominator by we would have to distribute it and we would end up with . We have not cleared the radical, only moved it to another part of the denominator. So our current method will not work. Instead we will use what is called a conjugate. A conjugate is made up of the same terms, with the opposite sign in the middle. So for our example with in the denominator, the conjugate would be
The advantage of a conjugate is when we multiply them together we have , which is a difference and a sum. We know when we multiply these, we get a difference of squares. Squaring and 5, with subtraction in the middle gives the product:
Our answer when multiplying conjugates will no longer have a square root, which is exactly what we want.
In the previous example, we could have reduced by dividing by -2 instead of 2, giving . Both answers are correct.
The same process can be used when there is a binomial in the numerator and denominator. We just need to remember to FOIL out the numerator.
Source: Adapted from "Beginning and Intermediate Algebra" by Tyler Wallace, an open source textbook available at: http://wallace.ccfaculty.org/book/book.html