Algebra

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# Rationalize the Denominator

#### Lesson Contents

## 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.

### Example Problems

**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}.**

**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}.**