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.
