<s>[INST] How to solve this radical expression I've been trying to solve this expression for at least two hours now... And I always get stuck towards the end, I don't know what I'm missing.
$\frac 1{xy} \times (\sqrt{xy} - \frac{xy}{x-\sqrt{xy}})\times (\sqrt{xy} + \frac{xy}{x+\sqrt{xy}})$
My first step was to rationalize the fractions inside the parenthesis like so
$\frac 1{xy} \times (\sqrt{xy} - \frac{xy(x+\sqrt{xy})}{(x-\sqrt{xy})(x+\sqrt{xy})})\times (\sqrt{xy} + \frac{xy(x-\sqrt{xy})}{(x+\sqrt{xy})(x-\sqrt{xy})})$
to get
$\frac 1{xy} \times (\sqrt{xy} - \frac{xy(x+\sqrt{xy})}{(x^2-xy)})\times (\sqrt{xy} + \frac{xy(x-\sqrt{xy})}{(x^2-xy)})$
then
$\frac 1{xy} \times (\sqrt{xy} - \frac{x^2y+ xy\sqrt{xy}}{(x^2-xy)})\times (\sqrt{xy} + \frac{x^2y-xy\sqrt{xy}}{(x^2-xy)})$
and then I'm kind of lost, nothing I've tried works.
I tried grouping each fraction by x and simplyfing removing it, and then computing the lcm inside the parenthesis in order to subtract the $\sqrt{xy}$. Or the other way around, first I did the lcm and subtracted and then simplyfied. I even tried multiplying the first factor by the second and simplifying as I went on. I tried using Wolfram Alpha to help me with each step, too. I think my calculations are correct, I'm just missing some simplification or something similar. The result should be
$\frac{x-4y}{x-y}$
I was able to get the $x-y$ but not the $x-4y$. I hope I didn't mess up the equations in the question, I'm really tired.
 [/INST] We have
$$\frac 1{xy} \times (\sqrt{xy} - \frac{xy}{x-\sqrt{xy}})\times (\sqrt{xy} + \frac{xy}{x+\sqrt{xy}})
=\frac 1{xy} \times \left(xy - \frac{x^2y^2}{x^2-xy}+\frac{xy\sqrt{xy}}{x+\sqrt{xy}}-\frac{xy\sqrt{xy}}{x-\sqrt{xy}}\right)=$$
$$=1 - \frac{xy}{x^2-xy}+\frac{\sqrt{xy}}{x+\sqrt{xy}}-\frac{\sqrt{xy}}{x-\sqrt{xy}}=$$
$$=1 - \frac{xy}{x^2-xy}+\frac{x\sqrt{xy}-xy}{x^2-xy}-\frac{x\sqrt{xy}+xy}{x^2-xy}=$$
$$=\frac{x^2-4xy}{x^2-xy}=\frac{x-4y}{x-y}$$
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