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

1. Problem Description

The problem is to simplify the expression:

$\sqrt[3]{48a^{12}b^6} \cdot \frac{ab}{\sqrt{a^2b^4}}$

2. Solution Steps

First, simplify the cube root term.

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

Next, simplify the square root term in the denominator.

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

Now, substitute the simplified expressions back into the original expression.

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

Now, simplify the fraction.

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

Substitute this back into the expression:

$2a^4b^2\sqrt[3]{6} \cdot \frac{1}{b}$

Finally, simplify the whole expression by canceling out $b$:

$2a^4b\sqrt[3]{6}$

3. Final Answer

The final answer is $2a^4b\sqrt[3]{6}$.