Constraints of $\frac{\sqrt{x^3}-\sqrt{y^3}+\sqrt{x^2y}-\sqrt{xy^2}}{\sqrt{x}+\sqrt{y}}$

Question

We have the expression A= $\frac{\sqrt{x^3}-\sqrt{y^3}+\sqrt{x^2y}-\sqrt{xy^2}}{\sqrt{x}+\sqrt{y}}$. I have to simplify it. First I want to define the constraints of $x$ and $y$ but I have some difficulties. The radicands must be $\ge 0$, thus $x^3 \ge 0, y^3 \ge 0, x^2y \ge 0, xy^2 \ge 0, x \ge 0, y \ge 0$. I am not sure about the last two inequalities. They are in the denominator so the sum of $\sqrt{x}$ and $\sqrt{y}$ mustn't be equal to $0$. If $x$ is equal to $0$, $y$ must be different ($y\ne0$).

I bought WolframAlpha two days ago, and I am trying to take advantage of it. I entered the expression, and I got two interesting things for me: 3D plot and contour plot. Can someone explain to me for what I can use them?
I also got this result: I suppose that this is the right answer but as you see 'assuming $x$ and $y$ are positive'. Can someone help me to solve the problem?

Answer 1

$$\frac{\left(\sqrt{x}-\sqrt{y}\right) \left(2 \sqrt{x y}+x+y\right)}{\sqrt{x}+\sqrt{y}}=\frac{\left(\sqrt{x}-\sqrt{y}\right) \left( \sqrt{x }+\sqrt y\right)^2}{\sqrt{x}+\sqrt{y}}=\left(\sqrt{x}-\sqrt{y}\right) \left( \sqrt{x }+\sqrt y\right)=x-y.$$

To be valid, require $x\ge0, y\ge0, x+y>0$.

Answer 2

The simplest form is:

$$\frac{\left(\sqrt{x}-\sqrt{y}\right) \left(2 \sqrt{x y}+x+y\right)}{\sqrt{x}+\sqrt{y}}$$

which you can get from Mathematica:

Assuming[x > 0 && y > 0,
 FullSimplify[
(Sqrt[x^3] - Sqrt[y^3] + Sqrt[x^2 y] - Sqrt[x y^2])/(Sqrt[x] + Sqrt[y])]
 ]

Answer 3

The expression needs $x$ and $y$ to be positive if you are using the regular square root whose domain is $[0,\infty)$. But that is not what WolframAlpha is doing, or else you would only see the graph plotted above the first quadrant. WolframAlpha is doing things like turning $\sqrt{-1}$ into $i$ and continuing with calculation.

For example, when $x=-1$ and $y=0$, your plot shows that WA is finding the output to be $1$. It's doing that this way: $$\begin{align} &\frac{\sqrt{(-1)^3}-\sqrt{0^3}+\sqrt{(-1)^2(0)}-\sqrt{(-1)0^2}}{\sqrt{-1}+\sqrt{0}}\\ &=\frac{\sqrt{-1}-\sqrt{0}+\sqrt{0}-\sqrt{0}}{\sqrt{-1}+\sqrt{0}}\\ &=\frac{i-0+0-0}{i+0}\\ &=\frac{i}{i}\\ &=1 \end{align}$$

Another way to say the above: WA is using a particular branch of the square root function with complex numbers as the domain, and cut in some way such that the square root of negative real numbers are "positive" imaginary.

In other answers here, it is shown how to simplify to $x-y$ when $x,y>0$. The simplification relies on facts about the regular square root that do not extend to this other square root WA is using.