Simplify cube root of 4x^2y* cube root of 16xy^2

$\sqrt[3]{4 x^{2} y} \cdot \sqrt[3]{16 x y^{2}}$

Rewrite $16 x y^{2}$ as $2^{3} \cdot (2 x y^{2})$ .

Factor $8$ out of $16$ .

$\sqrt[3]{4 x^{2} y} \cdot \sqrt[3]{8 (2) x y^{2}}$

Rewrite $8$ as $2^{3}$ .

$\sqrt[3]{4 x^{2} y} \cdot \sqrt[3]{2^{3} \cdot 2 x y^{2}}$

Add parentheses.

$\sqrt[3]{4 x^{2} y} \cdot \sqrt[3]{2^{3} \cdot 2 (x y^{2})}$

Add parentheses.

$\sqrt[3]{4 x^{2} y} \cdot \sqrt[3]{2^{3} \cdot (2 x y^{2})}$

Pull terms out from under the radical.

$\sqrt[3]{4 x^{2} y} \cdot (2 \sqrt[3]{2 x y^{2}})$

Rewrite using the commutative property of multiplication.

$2 \sqrt[3]{4 x^{2} y} \sqrt[3]{2 x y^{2}}$

Multiply $2 \sqrt[3]{4 x^{2} y} \sqrt[3]{2 x y^{2}}$ .

Combine using the product rule for radicals.

$2 \sqrt[3]{2 x y^{2} (4 x^{2} y)}$

Multiply $4$ by $2$ .

$2 \sqrt[3]{8 x y^{2} (x^{2} y)}$

Raise $x$ to the power of $1$ .

$2 \sqrt[3]{8 (x^{2} x^{1}) y^{2} y}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$2 \sqrt[3]{8 x^{2 + 1} y^{2} y}$

Add $2$ and $1$ .

$2 \sqrt[3]{8 x^{3} y^{2} y}$

Raise $y$ to the power of $1$ .

$2 \sqrt[3]{8 x^{3} (y^{1} y^{2})}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$2 \sqrt[3]{8 x^{3} y^{1 + 2}}$

Add $1$ and $2$ .

$2 \sqrt[3]{8 x^{3} y^{3}}$

Rewrite $8 x^{3} y^{3}$ as ${(2 x y)}^{3}$ .

$2 \sqrt[3]{{(2 x y)}^{3}}$

Pull terms out from under the radical, assuming real numbers.

$2 (2 x y)$

Multiply $2$ by $2$ .

$4 x y$