The problem asks to simplify the square root expression $\sqrt{192x^2y^3z^{12}}$.

1. Problem Description

The problem asks to simplify the square root expression $\sqrt{192x^2y^3z^{12}}$.

2. Solution Steps

First, we find the prime factorization of 192:

$192 = 2^6 \cdot 3$.

Then we can rewrite the expression as follows:

$\sqrt{192x^2y^3z^{12}} = \sqrt{2^6 \cdot 3 \cdot x^2 \cdot y^3 \cdot z^{12}}$.

Now, we take the square root of each factor:

$\sqrt{2^6} = 2^{6/2} = 2^3 = 8$

$\sqrt{3} = \sqrt{3}$

$\sqrt{x^2} = x$

$\sqrt{y^3} = \sqrt{y^2 \cdot y} = y\sqrt{y}$

$\sqrt{z^{12}} = z^{12/2} = z^6$

Putting it all together, we get:

$\sqrt{192x^2y^3z^{12}} = \sqrt{2^6} \cdot \sqrt{3} \cdot \sqrt{x^2} \cdot \sqrt{y^3} \cdot \sqrt{z^{12}} = 8 \cdot \sqrt{3} \cdot x \cdot y\sqrt{y} \cdot z^6 = 8xyz^6\sqrt{3y}$.

3. Final Answer

$8xyz^6\sqrt{3y}$