Simplifying Radical Expressions| Algebra| ENGG10 LE2 Problem 2

Rationalize the following expression by removing all radicals in the denominator.

\frac{3}{\sqrt{2}+\sqrt{5}-\sqrt{3}}

SOLUTION:

\frac{3}{\sqrt{2}+\sqrt{5}-\sqrt{3}}

=\frac{3}{\left(\sqrt{2}+\sqrt{5}\right)-\sqrt{3}}\cdot \frac{\left(\sqrt{2}+\sqrt{5}\right)+\sqrt{3}}{\left(\sqrt{2}+\sqrt{5}\right)+\sqrt{3}}

=\frac{3\left(\sqrt{2}+\sqrt{5}+\sqrt{3}\right)}{\left(\sqrt{2}+\sqrt{5}\right)^2-\left(\sqrt{3}\right)^2}

=\frac{3\left(\sqrt{2}+\sqrt{5}+\sqrt{3}\right)}{2+2\sqrt{10}+5-3}

=\frac{3\left(\sqrt{2}+\sqrt{5}+\sqrt{3}\right)}{4+2\sqrt{10}}

=\frac{3\left(\sqrt{2}+\sqrt{5}+\sqrt{3}\right)}{4+2\sqrt{10}}\cdot \frac{4-2\sqrt{10}}{4-2\sqrt{10}}

=\frac{3\left(\sqrt{2}+\sqrt{5}+\sqrt{3}\right)\left(4-2\sqrt{10}\right)}{\left(4\right)^2-\left(2\sqrt{10}\right)^2}

=\frac{3\left(\sqrt{2}+\sqrt{5}+\sqrt{3}\right)\left[2\left(2-\sqrt{10}\right)\right]}{16-40}

=\frac{6\left(\sqrt{2}+\sqrt{5}+\sqrt{3}\right)\left(2-\sqrt{10}\right)}{-24}

=-\frac{1}{4}\left(\sqrt{2}+\sqrt{5}+\sqrt{3}\right)\left(2-\sqrt{10}\right)

 

 

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