Simplify cube root of -9x^2y^4* cube root of 12xy

$\sqrt[3]{- 9 x^{2} y^{4}} \cdot \sqrt[3]{12 x y}$

Step 1

Rewrite $- 9 x^{2} y^{4}$ as ${(- y)}^{3} \cdot (9 x^{2} y)$ .

Rewrite $- 9$ as $- 1 (9)$ .

$\sqrt[3]{- 1 (9) x^{2} y^{4}} \cdot \sqrt[3]{12 x y}$

Rewrite $- 1$ as ${(- 1)}^{3}$ .

$\sqrt[3]{{(- 1)}^{3} \cdot 9 x^{2} y^{4}} \cdot \sqrt[3]{12 x y}$

Factor out $y^{3}$ .

$\sqrt[3]{{(- 1)}^{3} \cdot 9 x^{2} (y^{3} y)} \cdot \sqrt[3]{12 x y}$

Move $x^{2}$ .

$\sqrt[3]{{(- 1)}^{3} \cdot 9 y^{3} x^{2} y} \cdot \sqrt[3]{12 x y}$

Move $9$ .

$\sqrt[3]{{(- 1)}^{3} y^{3} \cdot 9 x^{2} y} \cdot \sqrt[3]{12 x y}$

Rewrite ${(- 1)}^{3} y^{3}$ as ${(- y)}^{3}$ .

$\sqrt[3]{{(- y)}^{3} \cdot 9 x^{2} y} \cdot \sqrt[3]{12 x y}$

Add parentheses.

$\sqrt[3]{{(- y)}^{3} \cdot 9 (x^{2} y)} \cdot \sqrt[3]{12 x y}$

Add parentheses.

$\sqrt[3]{{(- y)}^{3} \cdot (9 x^{2} y)} \cdot \sqrt[3]{12 x y}$

Step 2

Pull terms out from under the radical.

$- y \sqrt[3]{9 x^{2} y} \cdot \sqrt[3]{12 x y}$

Step 3

Multiply $- y \sqrt[3]{9 x^{2} y} \sqrt[3]{12 x y}$ .

Combine using the product rule for radicals.

$- y \sqrt[3]{12 x y (9 x^{2} y)}$

Multiply $9$ by $12$ .

$- y \sqrt[3]{108 x y (x^{2} y)}$

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

$- y \sqrt[3]{108 (x^{2} x^{1}) y \cdot y}$

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

$- y \sqrt[3]{108 x^{2 + 1} y \cdot y}$

Add $2$ and $1$ .

$- y \sqrt[3]{108 x^{3} y \cdot y}$

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

$- y \sqrt[3]{108 x^{3} (y^{1} y)}$

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

$- y \sqrt[3]{108 x^{3} (y^{1} y^{1})}$

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

$- y \sqrt[3]{108 x^{3} y^{1 + 1}}$

Add $1$ and $1$ .

$- y \sqrt[3]{108 x^{3} y^{2}}$

Step 4

Rewrite $108 x^{3} y^{2}$ as ${(3 x)}^{3} \cdot (4 y^{2})$ .

Factor $27$ out of $108$ .

$- y \sqrt[3]{27 (4) x^{3} y^{2}}$

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

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

Move $4$ .

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

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

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

Add parentheses.

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

Step 5

Pull terms out from under the radical.

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

Step 6

Simplify the expression.

Multiply $3$ by $- 1$ .

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

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

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

Simplify cube root of -9x^2y^4* cube root of 12xy

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