# 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.

• Should the denominator contain all of $x\sqrt{y}+y\sqrt{x}?$ Commented Jul 16, 2014 at 15:37
• Yes, that's the denominator. Commented Jul 16, 2014 at 15:44
• Ok, just asking because usually $a-b/c+d$ would be taken as $a-(b/c) +d.$ :) Commented Jul 16, 2014 at 15:47

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.

• ...can you not do a little more if you take $xy\neq0$? Commented Jul 16, 2014 at 15:50
• Thanks for the clear steps. Can I remove another xy and xy from the denominator of your final answer? So that I'm left with x-y? Commented Jul 16, 2014 at 15:58
• Yes, and this is because the original expression wasn't simplified to begin with (we could have cancelled $\sqrt{x}\sqrt{y}$ before we began anything else). Commented Jul 16, 2014 at 16:00
• Yes indeed, @user160137, provided $xy\neq 0$ we can divide through to get the last expression in above. Commented Jul 16, 2014 at 16:04
• @amWhy: But the original equation already prevents $xy=0$, so you do not need to make that extra claim... Commented Jul 16, 2014 at 16:14

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}

• I understand now. This is probably also what Maxim G said about it not beeing simplified from the beginning. Thanks! Commented Jul 16, 2014 at 16:05

The answer can be simplified even further (using the difference of two squares):

$$\frac{x-2\sqrt{xy} + y}{x-y} = \frac{(\sqrt{x} - \sqrt{y})^2}{(\sqrt{x} - \sqrt{y})(\sqrt{x} + \sqrt{y})} = \frac{1}{\sqrt{x} - \sqrt{y}} \text{ or } \frac{\sqrt{x} - \sqrt{y}}{x+y}$$