Why do we rationalize the denominator?
The most important reason, usually is "Because your teacher told you!"
There is nothing mathematically wrong with something like $$ \frac{3}{\red{ \sqrt{5}}} $$ or with any fraction that has a radical in its denominator
Just like in elementary school when you were probably told that fractions like $$\frac{7}{2} $$ 'bad' (or 'improper' in this case), having a radical in the denominator is not actually mathematically 'wrong'.
A historical reason: Before we had calculators that could compute radicals , we had to to calculate the value of radicals by hand, and it's much easier to do that when the radical is in the numerator.
Nowadays: There are two reasons why we still rationalize the denominator. Since most types of expressions and equations have a standard form, such a form was needed for rational expressions with radicals. And since we had historically rationalized the denominators due to a lack of calculators, this form became the standard one.
The good news: it's usually not very difficult to rationalize the denominator, and you can always double check your work with our calculator.
First, some background knowledge
In order to rationalize the denominator, you must multiply the numerator and denominator of a fraction by some radical that will make the 'radical' in the denominator go away.
Below is some background knowledge that you must remember in order to be able to understand the steps we are going to use.
Remember that you can create an equivalent fraction by multiplying your original fraction ($$ \frac{2}{3} $$ in the example below) by 1, $$ \red{\frac{5}{5} }$$ , or by $$ \frac{7}{7} $$ or by any form of 1 as a fraction
$ \frac{2}{3} \cdot \red{\frac{5}{5} } = \frac{10}{15} $
The product of two square roots is the square root of the products
(In other words you can multiply two square roots and put them under the same radical as shown below)$ \sqrt{2} \sqrt{3} = \sqrt{6} $
Full lesson on multiplying square roots.How to Rationalize the Denominator PowerPoint
Examples of rationalizing the denominator
Example 1 - Simplified Denominator
Rationalize the denominator of $$ \frac{2}{\sqrt{3}} $$
Note: this first example is the easiest type--It has a simplified denominator with no variables. Scroll down the page for more difficult examples
Step 1) Multiply the numerator and denominator of the original fraction $$ \left ( \frac{2}{\sqrt{3}} \right ) $$ by a number that will make the radical in the denominator 'go away' $$ \red{\frac{\sqrt{3}}{\sqrt{3}}}$$
Example 2 - Denominator not Simplified
Rationalize $$ \frac{3}{ 5 \sqrt{8} }$$
Step 1) Simplify the denominator
Step 3) Simplify
Example 3 - Conjugates
(more on rationalizing denominators with conjugates)Rationalize $$ \frac{3}{2 + \sqrt{5}} $$
Step 1) Simplify the square root in the denominator (if possible)
Step 3) Simplify
Practice Problems
This problem is like example 1 since its denominator is already simplified.
Step 1Multiply the numerator and denominator of the original fraction $$ \frac{3}{ \sqrt{7}} $$ by a number that will make the radical in the denominator 'go away' $$ \red{ \frac{\sqrt{7}}{ \sqrt{7}} } $$
This problem is like example 1 since its denominator is already simplified.
Step 1Multiply the numerator and denominator of the original fraction
$$ \frac{ 2}{3 \sqrt{11}} $$ by a number that will make the radical in the denominator 'go away' $$ \red{ \frac{\sqrt{11}}{ \sqrt{11}} } $$This problem is like example 2 since its denominator is not simplified.
Step 1Simplify the denominator
Multiply the numerator and denominator of the fraction
$$ \frac{5}{ 18\sqrt{3}} $$ by a number that will make the radical in the denominator 'go away' $$ \red{ \frac{\sqrt{3}}{ \sqrt{3}} } $$This problem is like example 2 since its denominator is not simplified.
Step 1Simplify the denominator
Multiply the numerator and denominator of the fraction
$$ \frac{ 3 \sqrt{2}}{ 14 \sqrt{3}} $$ by a number that will make the radical in the denominator 'go away' $$ \red{ \frac{\sqrt{3}}{ \sqrt{3}} } $$This problem is like example 2 since its denominator is not simplified.
Step 1Simplify the denominator
Multiply the numerator and denominator of the fraction
$$ \frac{ 5 \sqrt{6} }{ 6 \sqrt{3} } $$ by a number that will make the radical in the denominator 'go away' $$ \red{ \frac{\sqrt{3}}{ \sqrt{3}} } $$At first, this appears to be a problem like example 2 since its denominator is not simplified, but, if you look carefully, there's a quick way to solve this problem
Step 1simplify the denominator
Do you notice anything about the numerator and the denominator that could help us?
The $$ \red{\sqrt{21}} $$ from $$ \frac{ 11 \red{ \sqrt{21}} }{ 3 \sqrt{4} \sqrt{7} } $$ can be rewritten as $$ \sqrt{7} \sqrt{ 3} $$
Rationalize Denominator Widget
Simply type into the app below and edit the expression. The Math Way app will solve it form there. You can visit this calculator on its own page here.
To read our review of the Math way--which is what fuels this page's calculator, please go here.