Mathematic Problems
85. példa

Egyszerűsítse az alábbi kifejezést:

$\frac{\sqrt{x}+\sqrt{y}-1}{x+\sqrt{xy}}+\frac{\sqrt{x}-\sqrt{y}}{2\sqrt{xy}}\left(\frac{\sqrt{y}}{x-\sqrt{xy}}+\frac{\sqrt{y}}{x+\sqrt{xy}}\right)$

$\frac{\sqrt{x}+\sqrt{y}-1}{x+\sqrt{xy}}+\frac{\sqrt{x}-\sqrt{y}}{2\sqrt{xy}}\left(\frac{\sqrt{y}}{x-\sqrt{xy}}+\frac{\sqrt{y}}{x+\sqrt{xy}}\right)$
$=\frac{\sqrt{x}+\sqrt{y}-1}{x+\sqrt{xy}}+\frac{\sqrt{x}-\sqrt{y}}{2\sqrt{xy}}\left(\frac{\sqrt{y}\left(x+\sqrt{xy}\right)+\sqrt{y}\left(x-\sqrt{xy}\right)}{\left(x-\sqrt{xy}\right)\left(x+\sqrt{xy}\right)}\right)$

$=\frac{\sqrt{x}+\sqrt{y}-1}{x+\sqrt{xy}}+\frac{\sqrt{x}-\sqrt{y}}{2\sqrt{xy}}·\left[\frac{\sqrt{y}\left(x\overline{)+\sqrt{xy}}+x\overline{)-\sqrt{xy}}\right)}{\left(x-\sqrt{xy}\right)\left(x+\sqrt{xy}\right)}\right]$

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

$=\frac{\sqrt{x}+\sqrt{y}-1}{x+\sqrt{xy}}+\frac{\sqrt{x}-\sqrt{y}}{2\sqrt{xy}}\left(\frac{2\overline{)x}\sqrt{y}}{\overline{)x}\left(x-y\right)}\right)$

$=\frac{\sqrt{x}+\sqrt{y}-1}{x+\sqrt{xy}}+\frac{\sqrt{x}-\sqrt{y}}{2\sqrt{xy}}\left(\frac{2\sqrt{y}}{x-y}\right)$

$=\frac{\sqrt{x}+\sqrt{y}-1}{x+\sqrt{xy}}+\frac{\sqrt{x}-\sqrt{y}}{\overline{)2}\sqrt{x}\overline{)\sqrt{y}}}\left(\frac{\overline{)2\sqrt{y}}}{x-y}\right)$

$=\frac{\sqrt{x}+\sqrt{y}-1}{x+\sqrt{xy}}+\frac{\sqrt{x}-\sqrt{y}}{\sqrt{x}}\left(\frac{1}{x-y}\right)$
$=\frac{\sqrt{x}+\sqrt{y}-1}{x+\sqrt{xy}}+\frac{\sqrt{x}-\sqrt{y}}{\sqrt{x}}\left(\frac{1}{x-y}\right)$

$=\frac{\sqrt{x}+\sqrt{y}-1}{x+\sqrt{xy}}+\frac{\overline{)\sqrt{x}-\sqrt{y}}}{\sqrt{x}}·\frac{1}{\overline{)\left(\sqrt{x}-\sqrt{y}\right)}\left(\sqrt{x}+\sqrt{y}\right)}$

$=\frac{\sqrt{x}+\sqrt{y}-1}{x+\sqrt{xy}}+\frac{1}{\sqrt{x}}\text{}·\frac{1}{\sqrt{x}+\sqrt{y}}$
$=\frac{\sqrt{x}+\sqrt{y}-1}{x+\sqrt{xy}}+\frac{1}{\sqrt{x}}\text{}·\frac{1}{\sqrt{x}+\sqrt{y}}$
$=\frac{\sqrt{x}+\sqrt{y}-1}{x+\sqrt{xy}}+\frac{1}{x+\sqrt{xy}}$

$=\frac{\sqrt{x}+\sqrt{y}\overline{)-1}\overline{)+1}}{x+\sqrt{xy}}$

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

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

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

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

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

$=\frac{\sqrt{x}\overline{)\left(x-y\right)}}{x\overline{)\left(x-y\right)}}$

$=\frac{\sqrt{x}}{x}$