\(K=\dfrac{\left(\sqrt{x}-\sqrt{y}\right)^2+4\sqrt{xy}}{\sqrt{x}+\sqrt{y}}-\dfrac{x\sqrt{y}-y\sqrt{x}}{\sqrt{xy}}\)
\(=\dfrac{x-2\sqrt{xy}+y+4\sqrt{xy}}{\sqrt{x}+\sqrt{y}}-\dfrac{\sqrt{xy}\left(\sqrt{x}-\sqrt{y}\right)}{\sqrt{xy}}\)
\(=\dfrac{x+2\sqrt{xy}+y}{\sqrt{x}+\sqrt{y}}-\left(\sqrt{x}-\sqrt{y}\right)\)
\(=\dfrac{\left(\sqrt{x}+\sqrt{y}\right)^2}{\sqrt{x}+\sqrt{y}}-\left(\sqrt{x}-\sqrt{y}\right)\)
\(=\sqrt{x}+\sqrt{y}-\sqrt{x}+\sqrt{y}=2\sqrt{y}\)
\(I=\left(\dfrac{\sqrt{x}+\sqrt{y}}{\sqrt{x}-\sqrt{y}}-\dfrac{\sqrt{x}-\sqrt{y}}{\sqrt{x}+\sqrt{y}}\right):\dfrac{\sqrt{xy}}{x-y}\)
\(=\dfrac{\left(\sqrt{x}+\sqrt{y}\right)^2-\left(\sqrt{x}-\sqrt{y}\right)^2}{\left(\sqrt{x}+\sqrt{y}\right)\left(\sqrt{x}-\sqrt{y}\right)}\cdot\dfrac{x-y}{\sqrt{xy}}\)
\(=\dfrac{x+2\sqrt{xy}+y-x+2\sqrt{xy}-y}{x-y}\cdot\dfrac{x-y}{\sqrt{xy}}=\dfrac{4\sqrt{xy}}{\sqrt{xy}}=4\)
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