Rationalizing the Denominator

A. What It Means to Rationalize the Denominator

In order that all of us doing math can compare answers, we agree upon a common conversation, or set of rules, concerning the form of the answers.

For instance, we could easily agree that we would not leave an answer in the form of 3 + 4, but would write 7 instead.

When the topic switches to that of radicals, those doing math have agreed that a RADICAL IN SIMPLE FORM will not (among other things) have a radical in the denominator of a fraction. We will all change the form so there is no radical in the denominator.

Now a radical in the denominator will not be something as simple as . Instead, it will have a radicand which will not come out from under the radical sign like .

Since is an irrational number, and we need to make it NOT irrational, the process of changing its form so it is no longer irrational is called RATIONALIZING THE DENOMINATOR.

B. There are 3 Cases of Rationalizing the Denominator

1. Case I : There is ONE TERM in the denominator and it is a SQUARE ROOT.

2. Case II : There is ONE TERM in the denominator, however, THE INDEX IS GREATER THAN TWO. It might be a cube root or a fourth root.

3. Case III : There are TWO TERMS in the denominator.

Let's study Case I:

Example:

Procedure: Multiply top and bottom by the same radical.

one term square root Look at what is happening here!

Since squaring is the opposite of taking the square root, they cancel each other, leaving the 3