Rationalizing the denominator of a radical expression

Rationalizing a denominator is the process of removing the radical sign in the denominator of a radical expression.

Example #1:

$$ Rationalize \ \frac{3} {\sqrt{5}} $$
$$ Multiply \ by \ \frac{\sqrt{5}} {\sqrt{5}} $$

The reason we multiplied the denominator by square-root of 5 is because we want to make the denominator a perfect square. 

$$ Notice \ also \ that \ \frac{\sqrt{5}} {\sqrt{5}} = 1 $$

Therefore, it is like multiplying the expression by 1 which does not change the problem.

$$ \frac{3} {\sqrt{5}} = \frac{3} {\sqrt{5}} × \frac{\sqrt{5}} {\sqrt{5}} $$
$$ \frac{3} {\sqrt{5}} = \frac{3 \sqrt{5}} {\sqrt{25}} $$
$$ \frac{3} {\sqrt{5}} = \frac{3 \sqrt{5}} {5} $$


Example #2

$$ Rationalize \ \frac{ \sqrt{2}} {\sqrt{8n}} $$
$$ Multiply \ by \ \frac{\sqrt{2n}} {\sqrt{2n}} $$

Notice that if you had multiplied by square-root(8n), it will still be correct. Multiplying by square-root(2n) will give you smaller number to deal with though and that is better.

$$ \frac{\sqrt{2}} {\sqrt{8n}} = \frac{\sqrt{2}} {\sqrt{8n}} × \frac{\sqrt{2n}} {\sqrt{2n}} $$
$$ \frac{\sqrt{2}} {\sqrt{8n}} = \frac{\sqrt{2 × 2} × \sqrt{n}} {\sqrt{8n × 2n}}$$
$$ \frac{\sqrt{2}} {\sqrt{8n}} = \frac{\sqrt{4} × \sqrt{n}} {\sqrt{16n^2}}$$
$$ \frac{\sqrt{2}} {\sqrt{8n}} = \frac{2 × \sqrt{n}} {4n}$$
$$ \frac{\sqrt{2}} {\sqrt{8n}} = \frac{\sqrt{n}} {2n}$$


Rationalizing a denominator using conjugates

$$ Rationalize \ \frac{6} {\sqrt{5} - \sqrt{2}} $$
$$ Multiply \ by \ \ \frac{\sqrt{5} + \sqrt{2}} {\sqrt{5} + \sqrt{2}} $$
$$ \frac{6} {\sqrt{5} - \sqrt{2}} = \frac{6} {\sqrt{5} - \sqrt{2}} × \frac{\sqrt{5} + \sqrt{2}} {\sqrt{5} + \sqrt{2}} $$
$$ \frac{6} {\sqrt{5} - \sqrt{2}} = \frac{6 × (\sqrt{5} + \sqrt{2}) } { (\sqrt{5} - \sqrt{2})× (\sqrt{5} + \sqrt{2})} $$
$$ \frac{6} {\sqrt{5} - \sqrt{2}} = \frac{6 × (\sqrt{5} + \sqrt{2}) } { (\sqrt{5})^2 - (\sqrt{2})^2} $$
$$ \frac{6} {\sqrt{5} - \sqrt{2}} = \frac{6 × (\sqrt{5} + \sqrt{2}) } { 5 - 2} $$
$$ \frac{6} {\sqrt{5} - \sqrt{2}} = \frac{6 × (\sqrt{5} + \sqrt{2}) } { 2} $$
$$ \frac{6} {\sqrt{5} - \sqrt{2}} = 3 × (\sqrt{5} + \sqrt{2}) $$



Recent Articles

  1. How to Derive the Equation of an Ellipse Centered at the Origin

    Mar 13, 19 11:50 AM

    Learn how to derive the equation of an ellipse when the center of the ellipse is at the origin.

    Read More

New math lessons

Your email is safe with us. We will only use it to inform you about new math lessons.

            Follow me on Pinterest


Math quizzes

 Recommended

Scientific Notation Quiz

Graphing Slope Quiz

Adding and Subtracting Matrices Quiz  

Factoring Trinomials Quiz 

Solving Absolute Value Equations Quiz  

Order of Operations Quiz

Types of angles quiz


Tough algebra word problems

Tough Algebra Word Problems.

If you can solve these problems with no help, you must be a genius!

Recent Articles

  1. How to Derive the Equation of an Ellipse Centered at the Origin

    Mar 13, 19 11:50 AM

    Learn how to derive the equation of an ellipse when the center of the ellipse is at the origin.

    Read More

K-12 math tests


Everything you need to prepare for an important exam!

K-12 tests, GED math test, basic math tests, geometry tests, algebra tests. 

Real Life Math Skills

Learn about investing money, budgeting your money, paying taxes, mortgage loans, and even the math involved in playing baseball.