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

1. Problem Description

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

2. Solution Steps

We need to simplify the expression $\sqrt{54x^4y^7}$.

First, we can factor 54 into its prime factors: $54 = 2 \cdot 27 = 2 \cdot 3^3$.

So, the expression becomes $\sqrt{2 \cdot 3^3 x^4 y^7}$. We can rewrite this as

$\sqrt{2 \cdot 3^2 \cdot 3 \cdot x^4 \cdot y^6 \cdot y}$.

Now, we can take out the square roots of the perfect squares:

$\sqrt{3^2} = 3$

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

$\sqrt{y^6} = y^3$

So, the expression becomes:

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

Therefore, $\sqrt{54x^4y^7} = 3x^2y^3\sqrt{6y}$.

3. Final Answer

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