Represent the following mixed infinite decimal periodic fractions as common fractions:
$(\frac{1}{a - \sqrt{2}} - \frac{a^{2} + 4}{a^{3} - \sqrt{8}}) : {(\frac{a}{\sqrt{2}} + 1 + \frac{\sqrt{2}}{a})}^{- 1}$

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Solution

$(\frac{1}{a - \sqrt{2}} - \frac{a^{2} + 4}{a^{3} - \sqrt{8}}) : (\frac{a}{\sqrt{2}} + 1 + \frac{\sqrt{2}}{a})^{- 1}$
$= (\frac{a^{2} + \sqrt{2} a + 2 - a^{2} - 4}{a^{3} - \sqrt{8}}) \times (\frac{a^{2} + \sqrt{2} a + 2}{\sqrt{2} a})$
$= \frac{\sqrt{2} a - 2}{(a - \sqrt{2}) (a^{2} + \sqrt{2} a + 2)} \times \frac{(a^{2} + \sqrt{2} a + 2)}{\sqrt{2} a}$
$= \frac{\sqrt{2} (a - \sqrt{2})}{\sqrt{2} a (a - \sqrt{2})}$
$= \frac{1}{a}$

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