The problem asks us to simplify the expression $\sqrt[3]{48a^{12}b^6} \cdot \frac{ab}{\sqrt{a^2b^4}}$.

1. Problem Description

The problem asks us to simplify the expression $\sqrt[3]{48a^{12}b^6} \cdot \frac{ab}{\sqrt{a^2b^4}}$.

2. Solution Steps

First, we simplify the cube root.

$\sqrt[3]{48a^{12}b^6} = \sqrt[3]{8 \cdot 6 \cdot a^{12}b^6} = \sqrt[3]{2^3 \cdot 6 \cdot a^{12}b^6} = 2a^{12/3}b^{6/3}\sqrt[3]{6} = 2a^4b^2\sqrt[3]{6}$.

Next, we simplify the square root in the denominator.

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

Now we rewrite the expression:

$2a^4b^2\sqrt[3]{6} \cdot \frac{ab}{ab^2} = 2a^4b^2\sqrt[3]{6} \cdot \frac{a}{b^2} = 2a^5\sqrt[3]{6}$.

3. Final Answer

The final answer is $2a^5\sqrt[3]{6}$.