The given mathematical expression simplifies to:

$\frac{a \cdot \sqrt{a b} \cdot (a - b)}{2 \cdot \sqrt{b \cdot (a - 1)} \cdot (a + \sqrt{a b})}$

Let me know if you'd like a more detailed explanation, or if you have further questions!

Here are 5 related questions:

1. How can we further simplify the numerator or denominator of this expression?
2. What are the possible values for $a$ and $b$ to ensure the expression is defined?
3. How does the square root term in the expression influence the solution for $a$ and $b$?
4. Can this expression be factorized in any meaningful way?
5. What would happen to the expression if $b = 1$?

Tip: To simplify such expressions efficiently, identifying common factors and terms involving square roots early can help streamline the process.