$\dfrac{x-y}{\sqrt x -\sqrt y} - \dfrac{\sqrt{x^3} - \sqrt{y^3}}{x- y} (x\geq0 ;y\geq 0; x \neq y)$
$=\dfrac{(\sqrt x - \sqrt y)(\sqrt x +\sqrt y)}{\sqrt x - \sqrt y} - \dfrac{x\sqrt x - y \sqrt y}{(\sqrt x - \sqrt y)(\sqrt x + \sqrt y)}$
$=\sqrt x + \sqrt y - \dfrac{(\sqrt x - \sqrt y)(x - \sqrt{xy} + y)}{(\sqrt x - \sqrt x)(\sqrt x + \sqrt y)}$
$=\sqrt x +\sqrt y - \dfrac{x - \sqrt{xy} + y}{\sqrt x + \sqrt y}$
$=\dfrac{(\sqrt x + \sqrt y)^2 - x + \sqrt{xy} - y}{\sqrt x +\sqrt y}$
$=\dfrac{x + 2\sqrt{xy} + y - x + \sqrt{xy} - y}{\sqrt x + \sqrt y}$
$=\dfrac{3\sqrt{xy}}{\sqrt x +\sqrt y}$
d. $\bigg(\dfrac{x\sqrt x + y \sqrt y}{\sqrt x + \sqrt y} - \sqrt{xy}\bigg)\bigg(\dfrac{\sqrt x + \sqrt y}{x - y}\bigg)^2 (x>0;y>0;x \neq y)$
$=\bigg(\dfrac{(\sqrt x + \sqrt y)(x - \sqrt{xy} + y)}{\sqrt x + \sqrt y} - \sqrt{xy} \bigg)\bigg(\dfrac{\sqrt x + \sqrt x}{(\sqrt x + \sqrt y)(\sqrt x - \sqrt y)}\bigg)^2$
$=(x - \sqrt{xy} + y - \sqrt{xy})\bigg(\dfrac{1}{\sqrt x - \sqrt y}\bigg)^2$
$=(x - 2\sqrt{xy} +y)\dfrac{1}{(\sqrt x - \sqrt y)^2}$
$=(\sqrt x - \sqrt y)^2.\dfrac{1}{(\sqrt x - \sqrt y)^2}$
$= 1$
