Simplify the expression $3\sqrt{54x^4y^7z^6}$.

1. Problem Description

Simplify the expression $3\sqrt{54x^4y^7z^6}$.

2. Solution Steps

First, we simplify the number inside the square root:

$54 = 2 \cdot 27 = 2 \cdot 3^3 = 2 \cdot 3^2 \cdot 3 = 9 \cdot 6$.

Then, we consider the powers of $x$, $y$, and $z$:

$x^4 = (x^2)^2$

$y^7 = y^6 \cdot y = (y^3)^2 \cdot y$

$z^6 = (z^3)^2$

Now, we can rewrite the expression:

$3\sqrt{54x^4y^7z^6} = 3\sqrt{9 \cdot 6 \cdot x^4 \cdot y^6 \cdot y \cdot z^6} = 3\sqrt{3^2 \cdot 6 \cdot (x^2)^2 \cdot (y^3)^2 \cdot y \cdot (z^3)^2}$.

We can take out the perfect squares from under the square root:

$3\sqrt{3^2 \cdot (x^2)^2 \cdot (y^3)^2 \cdot (z^3)^2 \cdot 6y} = 3 \cdot 3 \cdot x^2 \cdot y^3 \cdot z^3 \sqrt{6y} = 9x^2y^3z^3\sqrt{6y}$.

3. Final Answer

$9x^2y^3z^3\sqrt{6y}$