How to Rationalize the Denominator

The denominator of a fraction is irrational if it contains a root. To rationalize the denominator, we need to multiply our fraction by another fraction that will cancel out the root in the denominator.

As an example, let's take a look at the irrational fraction ⁵/_√3. Since we can multiply anything by 1, we can multiply the fraction by ^√3/_√3. Doing so will cancel out the root in the denominator of the fraction, resulting in ^5√3/₃. The denominator of the fraction is now rational.

Definition of Rationalizing a Denominator

To rationalize something is to rid it of any irrational component. In algebra, a denominator is irrational if it has any roots. To rationalize the denominator is to remove the irrational component, which is the root/radical.

We rationalize the denominator because it is part of simplifying the fraction. Generally, we want a fraction in its simplest form before performing other algebraic operations on it. If a fraction has a root in the denominator, rationalizing it is a necessary step to reach its simplest form.

Rationalize the Denominator Example Problems

Let's go through a couple of example problems together to practice rationalizing the denominator.

Example Problem 1

Rationalize the denominator of ⁸/_√2.

Solution:

1. We will multiply the fraction by ^√2/_√2 to cancel out the radical in the denominator.
2. This gives us the rationalized fraction ^8√2/₂.

Example Problem 2

Rationalize the denominator of ⁷/_3√5.

Solution:

1. We will multiply the fraction by ^√5/_√5 to cancel out the radical in the denominator.
2. This gives us the rationalized fraction ^7√5/₁₅.