Algebra

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# Rationalize the Denominator

## 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.

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