Simplify the Square Root Fraction: √(2x²-4xy+2y²+(x-y)²)/(x-y)

To solve this problem, we'll follow these steps:

• Step 1: Simplify the expression inside the square root.
• Step 2: Utilize the algebraic identity to combine and simplify terms.
• Step 3: Divide the result by $x-y$.

Now, let's work through each step:

Step 1: Expand $(x-y)^2$.

We have $(x-y)^2 = x^2 - 2xy + y^2$.

Step 2: Simplify the square root expression:
The expression inside the square root is:
$2x^2 - 4xy + 2y^2 + (x-y)^2$.
Substitute $(x-y)^2 = x^2 - 2xy + y^2$:
$2x^2 - 4xy + 2y^2 + x^2 - 2xy + y^2$.

This simplifies to:
$3x^2 - 6xy + 3y^2$.

Notice that this can be rewritten using the identity $(a-b)^2 = a^2 - 2ab + b^2$ as:
$3(x^2 - 2xy + y^2) = 3(x-y)^2$.

Step 3: Extract the square root and simplify:
$\sqrt{3(x-y)^2} = \sqrt{3} \times |x-y|$.

Finally, divide by $x-y$:
$\frac{\sqrt{3} \times |x-y|}{x-y} = \sqrt{3} \times \frac{|x-y|}{x-y}$.

Since we assume $x-y \neq 0$, it simplifies to $\sqrt{3}$ because $\frac{|x-y|}{x-y} = 1$ when $x > y$, and $-1$ when $x < y$. With the absolute value, it remains $1$ in both cases.

Therefore, the solution to the problem is $\sqrt{3}$.