Simplify cube root of 3x^3y^2* cube root of 9xy^2

$\sqrt[3]{3 x^{3} y^{2}} \cdot \sqrt[3]{9 x y^{2}}$

Step 1

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

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

$\sqrt[3]{x^{3} \cdot 3 y^{2}} \cdot \sqrt[3]{9 x y^{2}}$

Add parentheses.

$\sqrt[3]{x^{3} \cdot (3 y^{2})} \cdot \sqrt[3]{9 x y^{2}}$

Step 2

Pull terms out from under the radical.

$x \sqrt[3]{3 y^{2}} \cdot \sqrt[3]{9 x y^{2}}$

Step 3

Multiply $x \sqrt[3]{3 y^{2}} \sqrt[3]{9 x y^{2}}$ .

Combine using the product rule for radicals.

$x \sqrt[3]{9 x y^{2} (3 y^{2})}$

Multiply $3$ by $9$ .

$x \sqrt[3]{27 x y^{2} y^{2}}$

Multiply $y^{2}$ by $y^{2}$ by adding the exponents.

Move $y^{2}$ .

$x \sqrt[3]{27 x (y^{2} y^{2})}$

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

$x \sqrt[3]{27 x y^{2 + 2}}$

Add $2$ and $2$ .

$x \sqrt[3]{27 x y^{4}}$

Step 4

Rewrite $27 x y^{4}$ as ${(3 y)}^{3} (x y)$ .

Rewrite $27$ as $3^{3}$ .

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

Factor out $y^{3}$ .

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

Move $x$ .

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

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

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

Add parentheses.

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

Step 5

Pull terms out from under the radical.

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

Step 6

Rewrite using the commutative property of multiplication.

$3 x y \sqrt[3]{x y}$