## Rationalize the Denominator Lesson

### 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:

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

#### Example Problem 2

Rationalize the denominator of ^{7}/_{3√5}.

Solution:

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