The given expression is:

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

To simplify this expression, we can break it down into smaller parts:

Step 1: Simplify the Numerator

The numerator is:

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

We will leave this as is for now because it's already simplified.

Step 2: Simplify the Denominator

The denominator is:

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

We can combine terms with $\frac{1}{\sqrt{xy}}$ as follows:

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

Since $\frac{x}{\sqrt{xy}} + \frac{y}{\sqrt{xy}} = \frac{x + y}{\sqrt{xy}}$, this simplifies to:

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

The terms $\frac{x + y}{\sqrt{xy}}$ cancel out, leaving:

$y$

Step 3: Simplify the Full Expression

Now we have:

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

This is the simplified form of the expression.

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Would you like to explore any further details or get more explanations? Here are some related questions:

1. How can we simplify expressions involving square roots further?
2. How do we combine terms involving radicals like $\sqrt{xy}$?
3. What methods can be used to simplify complex fractions?
4. How do we simplify complex algebraic expressions with multiple variables?
5. How can we handle expressions that have both linear terms and square roots?

Tip: Always check for common factors that can be simplified, especially when radicals are involved.