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

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

Step 1

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

Factor $8$ out of $16$ .

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

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

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

Add parentheses.

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

Add parentheses.

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

Step 2

Pull terms out from under the radical.

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

Step 3

Rewrite $4 x^{4} y$ as $x^{3} \cdot (4 x y)$ .

Factor out $x^{3}$ .

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

Reorder $4$ and $x^{3}$ .

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

Add parentheses.

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

Add parentheses.

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

Step 4

Pull terms out from under the radical.

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

Step 5

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

Combine using the product rule for radicals.

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

Multiply $4$ by $2$ .

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

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

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

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

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

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

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

Add $1$ and $1$ .

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

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

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

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

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

Add $1$ and $2$ .

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

Step 6

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

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

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

Move $x^{2}$ .

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

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

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

Step 7

Pull terms out from under the radical.

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

Step 8

Simplify the expression.

Rewrite using the commutative property of multiplication.

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

Multiply $2$ by $2$ .

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

Reorder factors in $4 x (y \sqrt[3]{x^{2}})$ .

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

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

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