The problem asks to simplify the expression $\sqrt[3]{54x^4y^2} \cdot \sqrt[3]{x^2y^4}$.

1. Problem Description

The problem asks to simplify the expression $\sqrt[3]{54x^4y^2} \cdot \sqrt[3]{x^2y^4}$.

2. Solution Steps

We can simplify this expression by using the property of radicals that $\sqrt[n]{a} \cdot \sqrt[n]{b} = \sqrt[n]{ab}$. Applying this property, we have

$\sqrt[3]{54x^4y^2} \cdot \sqrt[3]{x^2y^4} = \sqrt[3]{54x^4y^2 \cdot x^2y^4} = \sqrt[3]{54x^6y^6}$.

Now, we can simplify $\sqrt[3]{54}$ by factoring 54: $54 = 2 \cdot 27 = 2 \cdot 3^3$. So, $\sqrt[3]{54} = \sqrt[3]{2 \cdot 3^3} = 3\sqrt[3]{2}$.

Also, $\sqrt[3]{x^6} = x^{6/3} = x^2$ and $\sqrt[3]{y^6} = y^{6/3} = y^2$.

Therefore,

$\sqrt[3]{54x^6y^6} = \sqrt[3]{54} \cdot \sqrt[3]{x^6} \cdot \sqrt[3]{y^6} = 3\sqrt[3]{2} \cdot x^2 \cdot y^2 = 3x^2y^2\sqrt[3]{2}$.

3. Final Answer

The simplified expression is $3x^2y^2\sqrt[3]{2}$.