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

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Solution

$(\frac{a^{\frac{1}{2}} + 2}{a + 2 a^{\frac{1}{2}} + 1} - \frac{a^{\frac{1}{2}} - 2}{a - 1}) \cdot \frac{a^{\frac{1}{2}} + 1}{a^{\frac{1}{2}}}$
$= (\frac{\sqrt{a} + 2}{(\sqrt{a})^{2} + 2 \sqrt{a} + 1)} - \frac{\sqrt{a} - 2}{(\sqrt{a})^{2} - 1)}) \cdot \frac{\sqrt{a} + 1}{\sqrt{a}}$
$= [\frac{\sqrt{a} + 2}{(\sqrt{a} + 1)^{2}} - \frac{\sqrt{a} - 2}{(\sqrt{a} + 1) (\sqrt{a} - 1)}] \cdot \frac{\sqrt{a} + 1}{\sqrt{a}}$
$= [\frac{\sqrt{a} + 2}{\sqrt{a} + 1} - \frac{\sqrt{a} - 2}{\sqrt{a} - 1}] \frac{\sqrt{a} + 1}{\sqrt{a} (\sqrt{a} + 1)}$
$= (\frac{(\sqrt{a} + 2) (\sqrt{a} - 1) - (\sqrt{a} - 2) (\sqrt{a} + 1)}{a - 1}) \frac{1}{\sqrt{a}}$
$= [\frac{a - \sqrt{a} + 2 \sqrt{a} - 2 - (a + \sqrt{a} - 2 \sqrt{a} - 2)}{a - 1}] \frac{1}{\sqrt{a}}$
$= [\frac{a + \sqrt{a} - 2 - a + \sqrt{a} + 2}{a - 1}] \frac{1}{\sqrt{a}}$
$= \frac{2 \sqrt{a}}{(a - 1) \sqrt{a}} = \frac{2}{a - 1}$

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