Question

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

Answer 1

$(\frac{x^{\frac{1}{2}}+y^{\frac{1}{2}}}{x^{\frac{1}{2}}-y^{\frac{1}{2}}}-\frac{x^{\frac{1}{2}}-y^{\frac{1}{2}}}{x^{\frac{1}{2}}+y^{\frac{1}{2}}})(y^{\frac{-1}{2}}-x^{\frac{-1}{2}})$

$=\frac{(\sqrt{x}+\sqrt{y}{)}^2-(\sqrt{x}-\sqrt{y}{)}^2}{(\sqrt{x}{)}^2-(\sqrt{y}{)}^2}(\frac{1}{\sqrt{y}}-\frac{1}{\sqrt{x}})$

$=\frac{x+y+2\sqrt{x}y-x-y+2\sqrt{2}}{x-y}\times \frac{\sqrt{x}-\sqrt{y}}{\sqrt{xy}}$

$=\frac{4\sqrt{xy}(\sqrt{x}-\sqrt{y})}{(\sqrt{x}+\sqrt{y})(\sqrt{x}-\sqrt{y})\sqrt{xy}}$

$=\frac{4}{\sqrt{x}+\sqrt{y}}$