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
$$
Rationalize \ \frac{6} {\sqrt{5}  \sqrt{2}} $$


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


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


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


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


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


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


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



Jan 18, 19 10:53 AM
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