Simplify the expression $\sqrt[3]{-8a^3b^6} \times \sqrt[4]{16x^8y^{12}}$.

1. Problem Description

Simplify the expression $\sqrt[3]{-8a^3b^6} \times \sqrt[4]{16x^8y^{12}}$.

2. Solution Steps

First, we simplify the cube root:

$\sqrt[3]{-8a^3b^6} = \sqrt[3]{(-2)^3 a^3 (b^2)^3} = -2ab^2$

Next, we simplify the fourth root:

$\sqrt[4]{16x^8y^{12}} = \sqrt[4]{2^4 (x^2)^4 (y^3)^4} = 2x^2y^3$

Now, multiply the two simplified expressions:

$(-2ab^2) \times (2x^2y^3) = -4ab^2x^2y^3$

3. Final Answer

The final answer is $-4ab^2x^2y^3$.