<s>[INST] Simplify expression $(x\sqrt{y}- y\sqrt{x})/(x\sqrt{y} + y\sqrt{x})$ I'm stuck at the expression: $\displaystyle \frac{x\sqrt{y} -y\sqrt{x}}{x\sqrt{y} + y\sqrt{x}}$.
I need to simplify the expression (by making the denominators rational) and this is what I did:
$$(x\sqrt{y} - y\sqrt{x}) \times (x\sqrt{y} - y\sqrt{x})  =  (\sqrt{y} - \sqrt{x})^2$$
Divided by
$$(x\sqrt{y} + y\sqrt{x}) \times (x\sqrt{y} - y\sqrt{x} ) =  (x\sqrt{y})^2$$
So I'm left with $\displaystyle \frac{(\sqrt{y} - \sqrt{x})^2}{(x\sqrt{y})^2}$. 
This answer is incorrect. Can anyone help me understand what I did wrong? If there is a different approach to solve this it will also be much appreciated. Please explain in steps.
 [/INST] Given the question, we can write
\begin{eqnarray}
\frac{ x\sqrt{y} - y\sqrt{x} }{ x\sqrt{y} + y\sqrt{x} }
&=& \frac{ 1 - \sqrt{y/x} }{ 1 + \sqrt{y/x} }\\
&=& \frac{ \Big( 1 - \sqrt{y/x} \Big)^2 }
  { \Big( 1 + \sqrt{y/x} \Big) \Big( 1 - \sqrt{y/x} \Big)}\\
&=& \frac{ 1 + y/x - 2\sqrt{y/x} }{ 1 - y/x }\\
&=& \frac{ x + y - 2\sqrt{xy} }{ x - y }.
\end{eqnarray}

Or
\begin{eqnarray}
\frac{ x\sqrt{y} - y\sqrt{x} }{ x\sqrt{y} + y\sqrt{x} }
&=& \frac{ \sqrt{x} - \sqrt{y} }{ \sqrt{x} + \sqrt{y} }\\
&=& \frac{ \Big( \sqrt{x} - \sqrt{y} \Big)^2 }
  { \Big( \sqrt{x} + \sqrt{y} \Big) \Big( \sqrt{x} - \sqrt{y} \Big)}\\
&=& \frac{ x + y - 2\sqrt{xy} }{ x - y }.
\end{eqnarray}
</s>