Đáp án:
$\begin{array}{l}
\left( {\frac{{x\sqrt x + y\sqrt y }}{{\sqrt x + \sqrt y }} - \sqrt {xy} } \right).{\left( {\frac{{\sqrt x + \sqrt y }}{{x - y}}} \right)^2}\\
= \left[ {\frac{{{{\left( {\sqrt x } \right)}^3} + {{\left( {\sqrt y } \right)}^3}}}{{\sqrt x + \sqrt y }} - \sqrt {xy} } \right].{\left[ {\frac{{\sqrt x + \sqrt y }}{{\left( {\sqrt x - \sqrt y } \right)\left( {\sqrt x + \sqrt y } \right)}}} \right]^2}\\
= \left[ {\frac{{\left( {\sqrt x + \sqrt y } \right).\left( {x - \sqrt {xy} + y} \right)}}{{\sqrt x + \sqrt y }} - \sqrt {xy} } \right].\frac{1}{{{{\left( {\sqrt x - \sqrt y } \right)}^2}}}\\
= \left( {x - \sqrt {xy} + y - \sqrt {xy} } \right).\frac{1}{{{{\left( {\sqrt x - \sqrt y } \right)}^2}}}\\
= {\left( {\sqrt x - \sqrt y } \right)^2}.\frac{1}{{{{\left( {\sqrt x - \sqrt y } \right)}^2}}}\\
= 1\\
Vậy\,\left( {\frac{{x\sqrt x + y\sqrt y }}{{\sqrt x + \sqrt y }} - \sqrt {xy} } \right).{\left( {\frac{{\sqrt x + \sqrt y }}{{x - y}}} \right)^2} = 1
\end{array}$
