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.
Best Answer
For simplicity, set $z=\sqrt{xy}$, with $z^2=xy$.
Then your expression becomes \begin{align} \frac {1}{xy}\left(z - \frac{xy}{x-z}\right)\left(z + \frac{xy}{x+z}\right) &=\frac {1}{xy}\frac{xz-z^2-xy}{x-z}\frac{xz+z^2+xy}{x+z}\\[4px] &=\frac {1}{xy}\frac{xz-2xy}{x-z}\frac{xz+2xy}{x+z}\\[4px] &=\frac {1}{xy}\frac{x^2(z-2y)(z+2y)}{(x-z)(x+z)}\\[4px] &=\frac {1}{xy}\frac{x^2(z^2-4y^2)}{x^2-z^2}\\[4px] &=\frac {1}{y}\frac{x(xy-4y^2)}{x^2-xy}\\[4px] &=\frac {1}{y}\frac{xy(x-4y)}{x(x-y)}\\[4px] &=\frac{x-4y}{x-y} \end{align}
A similar strategy might be to set $a=\sqrt{x}$ and $b=\sqrt{y}$ (assuming both are positive, but the same holds when both are negative). Then we have $$ \sqrt{xy}-\frac{xy}{x-\sqrt{xy}}=ab-\frac{a^2b^2}{a^2-ab}=ab-\frac{ab^2}{a-b} =ab\left(1-\frac{b}{a-b}\right)=\frac{ab(a-2b)}{a-b} $$ Similarly $$ \sqrt{xy}+\frac{xy}{x+\sqrt{xy}}=\frac{ab(a+2b)}{a+b} $$ so your expression becomes $$ \frac{1}{a^2b^2}\frac{a^2b^2(a-2b)(a+2b)}{(a-b)(a+b)}= \frac{a^2-4b^2}{a^2-b^2}=\frac{x-4y}{x-y} $$