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√x-y = √x-√y   (???)    square both sides

x - y = x - 2√[x *y]  + y    subtract x from both sides  and rearrange

2√[x *y]  = 2y        divide both sides by 2

√[x * y]  = y

√x * √y  = y

√x * √y  - y  =  0        factor

√y [ √x  - √y]  = 0

So either    √y  = 0      or    √x  = √y

So....the original equation is true only when x = y   or    y = 0

NOOOOOO 

You can't 

it works for multiplcation or division 

If you mean $\sqrt{x-y}$, you cannot seperate $x$ and $y$ into $\sqrt{x}-\sqrt{y}$. If $x$ and $y$ was $\sqrt{x\times y}$, then you can seperate $x$ and $y$ into $\sqrt{x}\times\sqrt{y}$.  If $x$ and $y$ was $\sqrt\frac{{x}}{y}$, then you an seperate $x$ and $y$ into $\frac{{\sqrt x}}{\sqrt y}$.  If you mean $\sqrt{x}-y$, you cannot add a $\sqrt{}$ to y, that would change the problem entirely.

Example:

Let $x=3$ and let $y=2$

$\sqrt{x-y}$

$\sqrt{3-2}$

$\sqrt{1}$

$1$

The answer is $1$

Now lets do $\sqrt{x}-\sqrt{y}$

Let $x=3$ and let $y=2$

$\sqrt{x}-\sqrt{y}$

$\sqrt{3}-\sqrt{2}$

$1.7320508075688773...-1.414213562373095...$

$0.3178372451957823...$

The answer is $0.3178372451957823...$

$1$ and $0.3178372451957823...$ are not the same answer.

Now lets do $\sqrt{x\times y}$

Let $x=3$ and let $y=2$

$\sqrt{x\times y}$

$\sqrt{3\times 2}$

$\sqrt{6}$

$2.4494897427831781...$

The answer is $2.4494897427831781...$

Now lets do $\sqrt{x}\times\sqrt{y}$

Let $x=3$ and let $y=2$

$\sqrt{x}\times\sqrt{y}$

^{$\sqrt{3}\times\sqrt{2}$}

^{$1.7320508075688773... \times 1.414213562373095...$}

$2.4494897427831781...$

The answer is $2.4494897427831781...$

$2.4494897427831781...$ and $2.4494897427831781...$ are the same answer.

Now lets do $\sqrt\frac{{x}}{y}$

Let $x=3$ and let $y=2$

$\sqrt\frac{{x}}{y}$

$\sqrt\frac{{3}}{2}$

$1.224744871391589...$

The answer is $1.224744871391589...$

Now lets do $\frac{{\sqrt x}}{\sqrt y}$

Let $x=3$ and let $y=2$

$\frac{{\sqrt x}}{\sqrt y}$

$\frac{{\sqrt 3}}{\sqrt 2}$

$\frac{1.7320508075688773...}{1.414213562373095...}$

$1.224744871391589...$

The answer is $1.224744871391589...$

$1.224744871391589...$ and $1.224744871391589...$ is the same answer

Now lets do $\sqrt{x}-y$

Let $x=3$ and let $y=2$

$\sqrt{x}-y$

$\sqrt{3}-2$

$1.7320508075688773...-2$

$-0.2679491924311227...$

The answer is $-0.2679491924311227...$

Now let do $\sqrt{x}-\sqrt y$

Let $x=3$ and let $y=2$

$\sqrt{x}-\sqrt y$

$\sqrt{3}-\sqrt 2$

$1.7320508075688773...-1.414213562373095...$

$0.3178372451957823...$

The answer is $0.3178372451957823...$

$-0.2679491924311227...$ and $0.3178372451957823...$ are not the same answer

Hope this helps