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

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Solution

$\frac{x^{1 / 2} + x^{- 1 / 2}}{1 - x} + \frac{1 - x^{- 1 / 2}}{1 + \sqrt{x}} = \frac{1 + x}{\sqrt{x} (1 - x)} + \frac{\sqrt{x} - 1}{\sqrt{x} (\sqrt{x} + 1)} = \frac{1}{\sqrt{x}} (\frac{1 + x}{1 - x} - \frac{(\sqrt{x} - 1)^{2}}{1 - x}) = \frac{1 + x - x - 1 + 2 \sqrt{x}}{(1 - x) \sqrt{x}} = \frac{2}{1 - x}$

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