Rationalizing the Denominator

One of the “rules” for simplifying radicals is that you should never leave a radical in the denominator of a fraction. The reason for this rule is unclear (it appears to be a holdover from the days of slide rules), but it is nevertheless a rule that you will be expected to know in future math classes. The way to get rid of a square root is to multiply it by itself, which of course will give you whatever it was the square root of. To keep things legal, you must do to the numerator whatever you do to the denominator, and so we have the rule:

If the Denominator is Just a Single Radical

·         Multiply the numerator and denominator by the denominator

Example: 

·        Note: If you are dealing with an nth root instead of a square root, then you need n factors of that root in order to make it go away. For instance, if it is a cube root (n = 3), then you need to multiply by two more factors of that root to give a total of three factors.

If the Denominator Contains Two Terms

If the denominator contains a square root plus some other terms, a special trick does the job. It makes use of the difference of two squares formula:

            (a + b)(a – b) = a² – b²

Suppose that your denominator looked like a + b, where b was a square root and a represents all the other terms. If you multiply it by a – b, then you will end up with the square of your square root, which means no more square roots. It is called the conjugate when you replace the plus with a minus (or vice-versa). An example would help.

Example: