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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?

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

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 $$\frac{2 \sqrt{3}}{\sqrt{3}\sqrt{3}} = \frac{2 \sqrt{3}} { \sqrt{9}} = \frac{2 \sqrt{3}} { \color{red}{\sqrt{9}}} = \frac{2 \sqrt{3}} { \color{red}{3}}$$

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 $$\frac{3 \sqrt{2}}{ 10 \color{red}{\sqrt{2 }\sqrt{2} }} = \frac{3 \sqrt{2}}{ 10 \cdot \color{red}{2}} = \frac{3 \sqrt{2}}{ 20}$$

Example 3: 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 $$\frac{ 2}{3 \sqrt{11}}$$
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Problem 3) Rationalize $$\frac{5}{6 \sqrt{27}}$$
 Step 1

Problem 4) Rationalize $$\frac{ 3 \sqrt{2}}{ 7 \sqrt{12}}$$
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Problem 5) Rationalize $$\frac{ 5 \sqrt{6} }{ 2 \sqrt{27} }$$
 Step 1

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

## 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.