We need to simplify the expression: $\sqrt[3]{125a^9b^6} \cdot \frac{ab}{\sqrt{a^2b^4}}$.

1. Problem Description

We need to simplify the expression: $\sqrt[3]{125a^9b^6} \cdot \frac{ab}{\sqrt{a^2b^4}}$.

2. Solution Steps

First, simplify the cube root term:

$\sqrt[3]{125a^9b^6} = \sqrt[3]{5^3(a^3)^3(b^2)^3} = 5a^3b^2$

Next, simplify the square root term:

$\sqrt{a^2b^4} = \sqrt{a^2(b^2)^2} = ab^2$

Now, substitute the simplified terms back into the original expression:

$5a^3b^2 \cdot \frac{ab}{ab^2}$

Simplify the fraction:

$\frac{ab}{ab^2} = \frac{1}{b}$

Finally, multiply the simplified terms:

$5a^3b^2 \cdot \frac{1}{b} = \frac{5a^3b^2}{b} = 5a^3b$

3. Final Answer

$5a^3b$