algebra precalculus - Simplify expression $(xsqrt{y}- ysqrt{x})/(xsqrt{y} + ysqrt{x})$

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.

Answer

I am assuming your ambiguous notation begins with the task of simplifying:

$$\frac{x\sqrt y - y\sqrt x}{x\sqrt y + y\sqrt x}.$$

Assuming I'm correct, then we can rationalize the denominator (get rid of the factors with square roots), as follows:

Multiply the numerator and denominator by $(x\sqrt{y}-y\sqrt{x})$ to get a difference of squares. Recall that $$(a+b)(a-b) = a^2 - b^2.$$ If you carry out this multiplication, you'll have $$\dfrac{(x\sqrt{y}-y\sqrt{x})^2}{x^2y-xy^2}= \dfrac{x^2y - 2xy\sqrt{xy} + xy^2}{x^2y-xy^2}\; =\; \frac{xy(x-2\sqrt{xy} + y)}{xy(x-y)}\;= \; \frac{x-2\sqrt{xy} + y}{x-y}$$

You seemed to have the right idea, looking at your strategy, to multiply numerator and denominator by $x\sqrt y - y\sqrt x$, but you miscalculated.