Question

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

Answer 1

$\frac{a-b}{a+b+2\sqrt{ab}}:\frac{a^{\frac{-1}{2}}-b^{\frac{-1}{2}}}{a^{\frac{-1}{2}}+b^{\frac{-1}{2}}}$

$=\frac{(\sqrt{a}{)}^2-(\sqrt{b}{)}^2}{(\sqrt{a}{)}^2+(\sqrt{b}{)}^2+2\sqrt{a}\sqrt{b}}\times \frac{\frac{1}{\sqrt{a}}+\frac{1}{\sqrt{b}}}{\frac{1}{\sqrt{a}}-\frac{1}{\sqrt{b}}}$

$=\frac{(\sqrt{a}-\sqrt{b})(\sqrt{a}+\sqrt{b})}{(\sqrt{a}+\sqrt{b})(\sqrt{a}+\sqrt{b})}\times \frac{\frac{\sqrt{b}+\sqrt{a}}{\sqrt{ab}}}{\frac{\sqrt{b}-\sqrt{a}}{\sqrt{ab}}}$

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

$=-1$