Question

Represent the following mixed infinite decimal periodic fractions as common fractions:
$3(\frac{2}{\sqrt{10}+5}+\frac{5}{\sqrt{10}-2}-\frac{7}{\sqrt{10}})$

Answer 1

$3(\frac{2}{\sqrt{10}+5}+\frac{5}{\sqrt{10}-2}-\frac{7}{\sqrt{10}})$

$=3(\frac{2\times (\sqrt{10}-5)}{(\sqrt{10}+5)(\sqrt{10}-5)}+$$\frac{5\times (\sqrt{10}+2)}{(\sqrt{10}-2)(\sqrt{10}+2)}$ $+\frac{7\times \left(\sqrt{10}\right)}{\sqrt{10}\sqrt{10}})$ ....... (Rationalising)

$=3(\frac{2\sqrt{10}-10}{10-25}$ $+\frac{5\sqrt{10}+10}{10-4}$ $-\frac{7\sqrt{10}}{10})$

$=3(\frac{5\sqrt{10}+10}{6}-$ $\frac{2\sqrt{10}-10}{15}$ $-\frac{7\sqrt{10}}{10})$

$=3\sqrt{10}(\frac{5+\sqrt{10}}{6}$ - $\frac{2-\sqrt{10}}{15}$-$\frac{7}{10})$

$=3\sqrt{10}(\frac{25+5\sqrt{10}-4+2\sqrt{10}-21}{30})$

$=\frac{1}{\sqrt{10}}(2\sqrt{10}+5\sqrt{10})$

By taking $\sqrt{10}$ common

$=2+5=7$ is the answer.