Represent the following mixed infinite decimal periodic fractions as common fractions:
$(\frac{x^{1 / 2} + y^{1 / 2}}{x^{1 / 2} - y^{1 / 2}} - \frac{x^{1 / 2} - y^{1 / 2}}{x^{1 / 2} + y^{1 / 2}}) (y^{- 1 / 2} - x^{- 1 / 2})$

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Solution

$(\frac{x^{\frac{1}{2}} + y^{\frac{1}{2}}}{x^{\frac{1}{2}} - y^{\frac{1}{2}}} - \frac{x^{\frac{1}{2}} - y^{\frac{1}{2}}}{x^{\frac{1}{2}} + y^{\frac{1}{2}}}) (y^{\frac{- 1}{2}} - x^{\frac{- 1}{2}})$
$= \frac{(\sqrt{x} + \sqrt{y})^{2} - (\sqrt{x} - \sqrt{y})^{2}}{(\sqrt{x})^{2} - (\sqrt{y})^{2}} (\frac{1}{\sqrt{y}} - \frac{1}{\sqrt{x}})$
$= \frac{x + y + 2 \sqrt{x} y - x - y + 2 \sqrt{2}}{x - y} \times \frac{\sqrt{x} - \sqrt{y}}{\sqrt{x y}}$
$= \frac{4 \sqrt{x y} (\sqrt{x} - \sqrt{y})}{(\sqrt{x} + \sqrt{y}) (\sqrt{x} - \sqrt{y}) \sqrt{x y}}$
$= \frac{4}{\sqrt{x} + \sqrt{y}}$

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