Represent the following mixed infinite decimal periodic fractions as common fractions:
$(\frac{1}{m - \sqrt{m n}} + \frac{1}{m + \sqrt{m n}}) \cdot \frac{m^{3} - n^{3}}{m^{2} + m n + n^{2}}$

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Solution

$(\frac{1}{m - \sqrt{m n}} + \frac{1}{m + \sqrt{m n}})$ $\cdot \frac{m^{3} - n^{3}}{m^{2} + m n + n^{2}}$
$= (\frac{m + \sqrt{m n} + m - \sqrt{m n}}{m^{2} - (\sqrt{m n})^{2}}) \cdot \frac{(m - n) (m^{2} + m n + n^{2})}{(m^{2} + m n + n^{2})}$
$= \frac{2 m}{m^{2} - m n} \cdot (m - n)$
$= \frac{2 m (m - n)}{m (m - n)}$
$= 2$

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