$A=\dfrac{x\sqrt{x}+y\sqrt{y}}{\sqrt{x}+\sqrt{y}}-(\sqrt{x}-\sqrt{y})^2$
$=\dfrac{\sqrt{x}^3+\sqrt{y}^3}{\sqrt{x}+\sqrt{y}}-(x-2\sqrt{xy}+y)$
$=\dfrac{(\sqrt{x}+\sqrt{y})(x-\sqrt{xy}+y)}{\sqrt{x}+\sqrt{y}}-x+2\sqrt{xy}-y$
$=x-\sqrt{xy}+y-x+2\sqrt{xy}-y$
$=\sqrt{xy}$