Rationalizing the Denominator - Part One

Reference > Mathematics > Algebra > Simplifying Radicals

In the first section of this unit, one of the section questions was: Are

and

the same value?

The answer, it turns out, is "Yes, they are!" But they don't look the same, do they? Ideally, we should have a simplification rule that prevents us from having two answers that look so different, but have the same value. The rule is a fairly simple one: A radical expression is not in simplest form if it has a radical in its denominator.

Therefore,

is not in simplest form. That's good to know, but how do we get it into simplest form?

Easy! (No, really, it is quite easy!) Just multiply the numerator and the denominator by the square root of two. Watch what happens:

The rule is, whatever square root is in the denominator, you multiply both the numerator and denominator by that radical. Of course, it's a good idea to simplify the denominator as much as you can first. Consider the following example:

Simplify

512

The rule is, multiply the numerator and denominator by the radical, so we do this:

512

Notice that in one step we had to simplify

512

. We actually would have saved time if we'd done that at the beginning of the process:

512

Notice that by simplifying the denominator first, we only had to multiply numerator and denominator by

instead of

512

. Notice how many steps we saved by doing that! Always remember to simplify your denominators as much as possible before doing your multiplication.

But what if the radical in the denominator? What if the fraction was

First, we would rewrite the denominator in factored form:

2² · 3¹

The radical can only be removed if we make all the exponents inside the radical multiples of 3.

Thus, we need to multiply the top and bottom by

2¹ · 3²

. Watch what happens when we do that:

316

The same process works regardless of whether you're doing a cube root, a fourth root, or any other root. If you're doing an n^th root, figure out what you need to multiply each factor in the denominator by in order to turn the exponent into a multiple of n.