Question: Return the original user prompt in the "prompt" field. Wrap every mathematical expression, number used in a calculation, equation, fraction, variable, or formula in $...$ for inline math or $$...$$ for display math.
1. **State the problem:** Simplify the expression $$\sqrt{\frac{a - b}{ab}} \cdot \sqrt{\frac{a + ab}{a - b}} \cdot \sqrt{\frac{b - ab}{a + b}}.$$
2. **Combine the square roots:** Since all terms are multiplied under square roots, we can combine them into one square root:
$$\sqrt{\frac{a - b}{ab} \cdot \frac{a + ab}{a - b} \cdot \frac{b - ab}{a + b}}.$$
3. **Simplify inside the square root:** Cancel common factors where possible:
The factor $a - b$ appears in numerator and denominator:
$$\sqrt{\cancel{\frac{a - b}{ab}} \cdot \frac{a + ab}{\cancel{a - b}} \cdot \frac{b - ab}{a + b}} = \sqrt{\frac{a + ab}{ab} \cdot \frac{b - ab}{a + b}}.$$
4. **Rewrite terms:** Factor where possible:
$$a + ab = a(1 + b), \quad b - ab = b(1 - a).$$
So the expression becomes:
$$\sqrt{\frac{a(1 + b)}{ab} \cdot \frac{b(1 - a)}{a + b}} = \sqrt{\frac{a(1 + b)}{ab} \cdot \frac{b(1 - a)}{a + b}}.$$
5. **Simplify fractions:** Cancel $a$ and $b$ where possible:
$$\frac{a(1 + b)}{ab} = \frac{\cancel{a}(1 + b)}{\cancel{a}b} = \frac{1 + b}{b}.$$
6. **Substitute back:**
Expression inside the root is now:
$$\frac{1 + b}{b} \cdot \frac{b(1 - a)}{a + b}.$$
7. **Cancel $b$:**
$$\frac{1 + b}{\cancel{b}} \cdot \frac{\cancel{b}(1 - a)}{a + b} = \frac{(1 + b)(1 - a)}{a + b}.$$
8. **Final expression:**
$$\sqrt{\frac{(1 + b)(1 - a)}{a + b}}.$$
**Answer:** $$\boxed{\sqrt{\frac{(1 + b)(1 - a)}{a + b}}}.$$