Represent the following mixed infinite decimal periodic fractions as common fractions:
$\frac{a \sqrt{a} + b \sqrt{b}}{(\sqrt{a} + \sqrt{b}) (a - b)} + \frac{2 \sqrt{b}}{\sqrt{a} + \sqrt{b}} - \frac{\sqrt{a b}}{a - b}$

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Solution

$\frac{a \sqrt{a}}{(\sqrt{a} + \sqrt{b}) (a - b)} + \frac{2 \sqrt{b}}{\sqrt{a} + \sqrt{b}} - \frac{\sqrt{a b}}{a - b} = \frac{a \sqrt{a} + 2 a \sqrt{b} - 2 b \sqrt{b} - a \sqrt{b} - b \sqrt{a}}{(\sqrt{a} + \sqrt{b}) (a - b)} = \frac{a \sqrt{a} + a \sqrt{b} - b \sqrt{a} - 2 b \sqrt{b}}{(\sqrt{a} + \sqrt{b}) (a - b)}$

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