Question

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

Answer 1

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

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

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