Simplify the expression $\sqrt[3]{216x^4y^3}$.

1. Problem Description

Simplify the expression $\sqrt[3]{216x^4y^3}$.

2. Solution Steps

We are asked to simplify $\sqrt[3]{216x^4y^3}$. We can rewrite the expression as follows:

$\sqrt[3]{216x^4y^3} = \sqrt[3]{216} \cdot \sqrt[3]{x^4} \cdot \sqrt[3]{y^3}$

First, let's find the cube root of 216:

$216 = 6 \cdot 6 \cdot 6 = 6^3$

$\sqrt[3]{216} = \sqrt[3]{6^3} = 6$

Next, let's simplify $\sqrt[3]{x^4}$. We can rewrite $x^4$ as $x^3 \cdot x$.

$\sqrt[3]{x^4} = \sqrt[3]{x^3 \cdot x} = \sqrt[3]{x^3} \cdot \sqrt[3]{x} = x\sqrt[3]{x}$

Finally, let's simplify $\sqrt[3]{y^3}$.

$\sqrt[3]{y^3} = y$

Putting it all together:

$\sqrt[3]{216x^4y^3} = 6 \cdot x\sqrt[3]{x} \cdot y = 6xy\sqrt[3]{x}$

3. Final Answer

The final answer is $6xy\sqrt[3]{x}$