We are asked to simplify the expression $\sqrt{20x^4y^7z^6}$.

1. Problem Description

We are asked to simplify the expression $\sqrt{20x^4y^7z^6}$.

2. Solution Steps

First, we can rewrite $\sqrt{20x^4y^7z^6}$ as $\sqrt{20} \cdot \sqrt{x^4} \cdot \sqrt{y^7} \cdot \sqrt{z^6}$.

We simplify $\sqrt{20}$ as follows:

$\sqrt{20} = \sqrt{4 \cdot 5} = \sqrt{4} \cdot \sqrt{5} = 2\sqrt{5}$.

Next, we simplify $\sqrt{x^4}$:

$\sqrt{x^4} = x^{4/2} = x^2$.

Then, we simplify $\sqrt{y^7}$:

$\sqrt{y^7} = \sqrt{y^6 \cdot y} = \sqrt{y^6} \cdot \sqrt{y} = y^{6/2} \cdot \sqrt{y} = y^3\sqrt{y}$.

Finally, we simplify $\sqrt{z^6}$:

$\sqrt{z^6} = z^{6/2} = z^3$.

Putting it all together, we have:

$\sqrt{20x^4y^7z^6} = 2\sqrt{5} \cdot x^2 \cdot y^3\sqrt{y} \cdot z^3 = 2x^2y^3z^3\sqrt{5y}$.

3. Final Answer

$2x^2y^3z^3\sqrt{5y}$