Biến đổi vế trái, ta có:
$VT=\Bigg(\dfrac{\sqrt[]{x}}{\sqrt[]{x}-\sqrt[]{y}}-\dfrac{\sqrt[]{y}}{\sqrt[]{x}+\sqrt[]{y}}-\dfrac{2\sqrt[]{xy}}{x-y}\Bigg).(\sqrt[]{x}+\sqrt[]{y})$
$=\Bigg[\dfrac{\sqrt[]{x}}{\sqrt[]{x}-\sqrt[]{y}}-\dfrac{\sqrt[]{y}}{\sqrt[]{x}+\sqrt[]{y}}-\dfrac{2\sqrt[]{xy}}{(\sqrt[]{x}-\sqrt[]{y})(\sqrt[]{x}+\sqrt[]{y})}\Bigg].(\sqrt[]{x}+\sqrt[]{y})$
$=\dfrac{\sqrt[]{x}(\sqrt[]{x}+\sqrt[]{y})-\sqrt[]{y}(\sqrt[]{x}-\sqrt[]{y})-2\sqrt[]{xy}}{(\sqrt[]{x}-\sqrt[]{y})(\sqrt[]{x}+\sqrt[]{y})}.(\sqrt[]{x}+\sqrt[]{y})$
$=\dfrac{x+\sqrt[]{xy}-\sqrt[]{xy}+y-2\sqrt[]{xy}}{(\sqrt[]{x}+\sqrt[]{y})(\sqrt[]{x}-\sqrt[]{y})}.(\sqrt[]{x}+\sqrt[]{y})$
$=\dfrac{x-2\sqrt[]{xy}+y}{\sqrt[]{x}-\sqrt[]{y}}$
$=\dfrac{(\sqrt[]{x}-\sqrt[]{y})^2}{\sqrt[]{x}-\sqrt[]{y}}$
$=\sqrt[]{x}-\sqrt[]{y}$
$=VP$ (đpcm)
