Simplify 4y cube root of 81x^7y^7

$4 y \sqrt[3]{81 x^{7} y^{7}}$

Step 1

Rewrite $81 x^{7} y^{7}$ as ${(3 x^{2} y^{2})}^{3} \cdot (3 x y)$ .

Factor $27$ out of $81$ .

$4 y \sqrt[3]{27 (3) x^{7} y^{7}}$

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

$4 y \sqrt[3]{3^{3} \cdot 3 x^{7} y^{7}}$

Factor out $x^{6}$ .

$4 y \sqrt[3]{3^{3} \cdot 3 (x^{6} x) y^{7}}$

Rewrite $x^{6}$ as ${(x^{2})}^{3}$ .

$4 y \sqrt[3]{3^{3} \cdot 3 ({(x^{2})}^{3} x) y^{7}}$

Factor out $y^{6}$ .

$4 y \sqrt[3]{3^{3} \cdot 3 ({(x^{2})}^{3} x) (y^{6} y)}$

Rewrite $y^{6}$ as ${(y^{2})}^{3}$ .

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

Move $x$ .

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

Move $3$ .

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

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

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

Add parentheses.

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

Add parentheses.

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

Step 2

Pull terms out from under the radical.

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

Step 3

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

Move $y^{2}$ .

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

Multiply $y^{2}$ by $y$ .

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

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

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

$4 y^{2 + 1} (3 x^{2} \sqrt[3]{3 x y})$

Add $2$ and $1$ .

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

Step 4

Multiply $3$ by $4$ .

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

Step 5

Reorder factors in $12 y^{3} (x^{2} \sqrt[3]{3 x y})$ .

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