The expression $$\frac{x\left(\frac{\sqrt{x}+\sqrt{y}}{2y\sqrt{x}}\right)^{-1}+y\left(\frac{\sqrt{x}+\sqrt{y}}{2x\sqrt{y}}\right)^{-1}}{\left(\frac{x+\sqrt{xy}}{2xy}\right)^{-1}+\left(\frac{y+\sqrt{xy}}{2xy}\right)^{-1}}$$ reduces to
Solution
The numerator becomes $$\frac{2xy\sqrt{x}}{\sqrt{x}+\sqrt{y}}+\frac{2xy\sqrt{y}}{\sqrt{x}+\sqrt{y}}=2xy$$. Writing $$x=\left(\sqrt{x}\right)^2$$ and $$y=\left(\sqrt{y}\right)^2$$, the denominator becomes $$\frac{2xy}{x+\sqrt{xy}}+\frac{2xy}{y+\sqrt{xy}}=2\sqrt{xy}$$. Therefore the expression equals $$\frac{2xy}{2\sqrt{xy}}=\sqrt{xy}$$.
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