Question

Represent the following mixed infinite decimal periodic fractions as common fractions:
$(\frac{1}{a-\sqrt{2}}-\frac{a^2+4}{a^3-\sqrt{8}}):{(\frac{a}{\sqrt{2}}+1+\frac{\sqrt{2}}{a})}^{-1}$

Answer 1

$(\frac{1}{a-\sqrt{2}}-\frac{a^2+4}{a^3-\sqrt{8}}):(\frac{a}{\sqrt{2}}+1+\frac{\sqrt{2}}{a}{)}^{-1}$

$=\left(\frac{a^2+\sqrt{2}a+2-a^2-4}{a^3-\sqrt{8}}\right)\times \left(\frac{a^2+\sqrt{2}a+2}{\sqrt{2}a}\right)$

$=\frac{\sqrt{2}a-2}{(a-\sqrt{2})(a^2+\sqrt{2}a+2)}\times \frac{(a^2+\sqrt{2}a+2)}{\sqrt{2}a}$

$=\frac{\sqrt{2}(a-\sqrt{2})}{\sqrt{2}a(a-\sqrt{2})}$

$=\frac{1}{a}$