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 squareroot 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 squareroot(8n), it will still be correct. Multiplying by squareroot(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
$$
Rationalize \ \frac{6} {\sqrt{5}  \sqrt{2}} $$


$$
Multiply \ by \ \ \frac{\sqrt{5} + \sqrt{2}} {\sqrt{5} + \sqrt{2}} $$


$$
\frac{6} {\sqrt{5}  \sqrt{2}} =
\frac{6} {\sqrt{5}  \sqrt{2}} ×
\frac{\sqrt{5} + \sqrt{2}} {\sqrt{5} + \sqrt{2}} $$


$$
\frac{6} {\sqrt{5}  \sqrt{2}} =
\frac{6 × (\sqrt{5} + \sqrt{2}) } { (\sqrt{5}  \sqrt{2})× (\sqrt{5} + \sqrt{2})} $$


$$
\frac{6} {\sqrt{5}  \sqrt{2}} =
\frac{6 × (\sqrt{5} + \sqrt{2}) } { (\sqrt{5})^2  (\sqrt{2})^2} $$


$$
\frac{6} {\sqrt{5}  \sqrt{2}} =
\frac{6 × (\sqrt{5} + \sqrt{2}) } { 5  2} $$


$$
\frac{6} {\sqrt{5}  \sqrt{2}} =
\frac{6 × (\sqrt{5} + \sqrt{2}) } { 2} $$


$$
\frac{6} {\sqrt{5}  \sqrt{2}} =
3 × (\sqrt{5} + \sqrt{2}) $$



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.