Đáp án:
$\begin{array}{l}
M = \left( {\dfrac{{\sqrt x + \sqrt y }}{{2\left( {\sqrt x - \sqrt y } \right)}} - \dfrac{{\sqrt x - \sqrt y }}{{2\left( {\sqrt x - \sqrt y } \right)}} + \dfrac{{xy}}{{x - y}}} \right):\dfrac{y}{{x - y}}\\
= \left( {\dfrac{{\sqrt x + \sqrt y - \sqrt x + \sqrt y }}{{2\left( {\sqrt x - \sqrt y } \right)}} + \dfrac{{xy}}{{\left( {\sqrt x - \sqrt y } \right)\left( {\sqrt x + \sqrt y } \right)}}} \right)\\
.\dfrac{{x - y}}{y}\\
= \left( {\dfrac{{2\sqrt y }}{{2\left( {\sqrt x - \sqrt y } \right)}} + \dfrac{{xy}}{{\left( {\sqrt x - \sqrt y } \right).\left( {\sqrt x + \sqrt y } \right)}}} \right).\dfrac{{x - y}}{y}\\
= \left( {\dfrac{{\sqrt y \left( {\sqrt x + \sqrt y } \right) + xy}}{{\left( {\sqrt x - \sqrt y } \right).\left( {\sqrt x + \sqrt y } \right)}}} \right).\dfrac{{\left( {\sqrt x - \sqrt y } \right)\left( {\sqrt x + \sqrt y } \right)}}{y}\\
= \dfrac{{\sqrt {xy} + y + xy}}{y}\\
= \dfrac{{\sqrt x + \sqrt y + x\sqrt y }}{{\sqrt y }}\\
= \dfrac{{\sqrt x }}{{\sqrt y }} + 1 + x\\
N = \left( {\dfrac{1}{{1 - \sqrt x }} - 1} \right).\left( {\sqrt x - \dfrac{{1 - 2\sqrt x }}{{1 - \sqrt x }} + 1} \right)\\
= \dfrac{{1 - 1 + \sqrt x }}{{1 - \sqrt x }}:\dfrac{{\sqrt x .\left( {1 - \sqrt x } \right) - 1 + 2\sqrt x + 1 - \sqrt x }}{{1 - \sqrt x }}\\
= \dfrac{{\sqrt x }}{{1 - \sqrt x }}.\dfrac{{1 - \sqrt x }}{{\sqrt x - x - 1 + 2\sqrt x + 1 - \sqrt x }}\\
= \dfrac{{\sqrt x }}{{ - x + 2\sqrt x }}\\
= \dfrac{1}{{2 - \sqrt x }}
\end{array}$
