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How to Rationalize the Denominator

Examples, formula and the Steps!

Rationalize the Denominator Worksheet
(25 question worksheet with answer key)
Radicals Home
$$ \sqrt{45 } = \text{?} $$
Rationalize denominators with Conjugates

Why do we rationalize the denominator?

question
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.
Multiplying a fraction by 1 Multiply Radicals
Remember that you can create an equivalent fraction by multiplying your original fraction ($$ \frac{2}{3} $$ in the example below) by 1 , $$ \color{red}{\frac{5}{5} }$$ , or by $$ \frac{7}{7} $$ or by any form of 1 as a fraction
$ \frac{2}{3} \cdot \color{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


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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' $$ \color{red}{\frac{\sqrt{3}}{\sqrt{3}}} $$ $$ \frac{2}{\sqrt{3}} \cdot \color{red}{\frac{\sqrt{3}}{\sqrt{3}}} $$
Step 3) Simplify $$ \frac{2 \sqrt{3}}{\sqrt{3}\sqrt{3}} = \frac{2 \sqrt{3}} { \sqrt{9}} = \frac{2 \sqrt{3}} { \color{red}{\sqrt{9}}} = \frac{2 \sqrt{3}} { \color{red}{3}} $$

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Example 2: Denominator not Simplified


Rationalize $$ \frac{3}{ 5 \sqrt{8} } $$
Step 1) Simplify the denominator $$ \frac{3}{ 5 \color{red}{\sqrt{8}} } = \frac{3}{ 5 \color{red}{\sqrt{4} \sqrt{2}} } = \frac{3}{ 5 \cdot \color{red}{2 }\sqrt{2} } = \frac{3}{ 10 \sqrt{2} } $$
Step 2) Multiply the numerator and denominator by a number that will make the radical in the denominator 'go away' : $$ \color{red}{ \frac{ \sqrt{2} }{\sqrt{2}}} $$ $$ \frac{3}{ 10 \sqrt{2} } \cdot \color{red}{ \frac{ \sqrt{2} }{\sqrt{2}}} $$
Step 2) Simplify $$ \frac{3 \sqrt{2}}{ 10 \color{red}{\sqrt{2 }\sqrt{2} }} = \frac{3 \sqrt{2}}{ 10 \cdot \color{red}{2}} = \frac{3 \sqrt{2}}{ 20} $$

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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) $$ \sqrt{5} $$ is already simplified, so nothing to do here
Step 2) Multiply the numerator and denominator by the denominator's conjugate $$ 2 \color{red}{-} \sqrt{5} $$ $$ \frac{3}{2 + \sqrt{5}} \color{red}{ \frac{2 - \sqrt{5}} {2 - \sqrt{5}}} = \frac{3 (2 \color{red}{-} \sqrt{5} )} { (2\color{red}{+} \sqrt{5} )(2\color{red}{-} \sqrt{5} ) } \\ = \frac{ 6 -3\sqrt{5} } { 4 \color{red}{+} 2\sqrt{5} \color{red}{-} 2\sqrt{5}+ 5 } = \frac{ 6 -3\sqrt{5} } { 4 \cancel { \color{red}{+}2\sqrt{5} \color{red}{-} 2\sqrt{5} } + 5 } \\ = \frac{ 6 -3\sqrt{5} } { 4 + 5 } = \frac{ 6 -3\sqrt{5} } { 9 } $$
Step 3) Simplify $$ \frac{ 6 -3\sqrt{5} } { 9 } = \frac{ \color{red}{3}(2 -1\sqrt{5}) } { \color{red}{3}(3) } \\ \frac{ \color{red}{\cancel{3}}(2 -1\sqrt{5} ) } { \color{red}{\cancel{3} } (3 ) } = \frac{ (2 - \sqrt{5} )} { 3 } $$
Practice Problems
Problem 1) Rationalize $$ \frac{3}{ \sqrt{7}} $$
Step 1


Problem 2) Rationalize $$ \frac{ 2}{3 \sqrt{11}} $$
Step 1


Problem 3) Rationalize $$ \frac{5}{6 \sqrt{27}} $$
Step 1


Problem 4) Rationalize $$ \frac{ 3 \sqrt{2}}{ 7 \sqrt{12}} $$
Step 1

Problem 5) Rationalize $$ \frac{ 5 \sqrt{6} }{ 2 \sqrt{27} } $$
Step 1

Problem 6) Rationalize $$ \frac{ 11 \sqrt{21} }{ 3 \sqrt{28} } $$
Step 1

This problem is like example 2 since its denominator is not simplified.
Step 1) Simplify the denominator
Step 2)Multiply the numerator and denominator by a number that will make the radical in the denominator 'go away'
Step 3) Simplify

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.