Simplify 2y cube root of 27x^5y^7

$2 y \sqrt[3]{27 x^{5} y^{7}}$

Rewrite $27 x^{5} y^{7}$ as ${(3 x y^{2})}^{3} (x^{2} y)$ .

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

$2 y \sqrt[3]{3^{3} x^{5} y^{7}}$

Factor out $x^{3}$ .

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

Factor out $y^{6}$ .

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

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

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

Move $x^{2}$ .

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

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

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

Add parentheses.

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

Pull terms out from under the radical.

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

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

Move $y^{2}$ .

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

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

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

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

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

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

Add $2$ and $1$ .

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

Multiply $3$ by $2$ .

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

Reorder factors in $6 y^{3} (x \sqrt[3]{x^{2} y})$ .

$6 y^{3} x \sqrt[3]{x^{2} y}$