Represent the following mixed infinite decimal periodic fractions as common fractions:
$\frac{\frac{x - y}{\sqrt{x} - \sqrt{y}} - \frac{x - y}{\sqrt{x} + \sqrt{y}}}{\frac{\sqrt{x} - \sqrt{y}}{x - y} + \frac{\sqrt{x} + \sqrt{y}}{x - y}} \cdot \frac{2 \sqrt{x y}}{y - x}$

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Solution

$\frac{x - y}{\sqrt{x} - \sqrt{y}} - \frac{x - y}{\sqrt{x} + \sqrt{y}} = (x - y) \frac{2 \sqrt{y}}{x - y} = 2 \sqrt{y}$
$\frac{\sqrt{x} - \sqrt{y}}{x - y} + \frac{\sqrt{x} + \sqrt{y}}{x - y} = \frac{2 \sqrt{x}}{x - y}$
$\frac{\frac{x - y}{\sqrt{x} - \sqrt{y}} - \frac{x - y}{\sqrt{x} + \sqrt{y}}}{\frac{\sqrt{x} - \sqrt{y}}{x - y} + \frac{\sqrt{x} + \sqrt{y}}{x - y}} . \frac{2 \sqrt{x} \sqrt{y}}{y - x} = - 2 y$

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