\(VT=\left(\dfrac{x^{\dfrac{1}{2}}-y^{\dfrac{1}{2}}}{xy^{\dfrac{1}{2}}+x^{\dfrac{1}{2}}y}+\dfrac{x^{\dfrac{1}{2}}+y^{\dfrac{1}{2}}}{xy^{\dfrac{1}{2}}-x^{\dfrac{1}{2}}y}\right)\cdot\dfrac{x^{\dfrac{3}{2}}y^{\dfrac{1}{2}}}{x+y}-\dfrac{2y}{x-y}\)
\(=\left(\dfrac{\sqrt{x}-\sqrt{y}}{x\sqrt{y}+y\sqrt{x}}+\dfrac{\sqrt{x}+\sqrt{y}}{x\sqrt{y}-y\sqrt{x}}\right)\cdot\dfrac{x^{\dfrac{1}{2}}y^{\dfrac{1}{2}}\cdot x}{x+y}-\dfrac{2y}{x-y}\)
\(=\left(\dfrac{\sqrt{x}-\sqrt{y}}{\sqrt{xy}\left(\sqrt{x}+\sqrt{y}\right)}+\dfrac{\sqrt{x}+\sqrt{y}}{\sqrt{xy}\left(\sqrt{x}-\sqrt{y}\right)}\right)\cdot\dfrac{\sqrt{xy}\cdot x}{x+y}-\dfrac{2y}{x-y}\)
\(=\dfrac{\left(\sqrt{x}-\sqrt{y}\right)^2+\left(\sqrt{x}+\sqrt{y}\right)^2}{\sqrt{xy}\left(\sqrt{x}-\sqrt{y}\right)\left(\sqrt{x}+y\right)}\cdot\dfrac{x\sqrt{xy}}{x+y}-\dfrac{2y}{x-y}\)
\(=\dfrac{2\left(x+y\right)}{x-y}\cdot\dfrac{x}{x+y}-\dfrac{2y}{x-y}\)
\(=\dfrac{2x\left(x+y\right)-2y\left(x+y\right)}{\left(x-y\right)\left(x+y\right)}=\dfrac{2x^2+2xy-2xy-2y^2}{x^2-y^2}\)
\(=\dfrac{2x^2-2y^2}{x^2-y^2}=2\)
