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

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Solution

$(\frac{\sqrt{a}}{2} - \frac{1}{2 \sqrt{a}})^{2} (\frac{\sqrt{a} - 1}{\sqrt{a} + 1} - \frac{\sqrt{a} + 1}{\sqrt{a} - 1})$
$= (\frac{a - 1}{2 \sqrt{a}})^{2} [\frac{(\sqrt{a} - 1)^{2} - (\sqrt{a} + 1)^{2}}{a - 1}]$
$= \frac{(a - 1)^{2}}{4 a} \frac{- 4 a}{(a - 1)}$
$= (- a + 1)$

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