Simplify 4y^3 cube root of 81x^8y^8

$4 y^{3} \sqrt[3]{81 x^{8} y^{8}}$

Rewrite $81 x^{8} y^{8}$ as ${(3 x^{2} y^{2})}^{3} \cdot (3 x^{2} y^{2})$ .

Factor $27$ out of $81$ .

$4 y^{3} \sqrt[3]{27 (3) x^{8} y^{8}}$

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

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

Factor out $x^{6}$ .

$4 y^{3} \sqrt[3]{3^{3} \cdot 3 (x^{6} x^{2}) y^{8}}$

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

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

Factor out $y^{6}$ .

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

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

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

Move $x^{2}$ .

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

Move $3$ .

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

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

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

Add parentheses.

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

Add parentheses.

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

Pull terms out from under the radical.

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

Multiply $y^{3}$ by $y^{2}$ by adding the exponents.

Move $y^{2}$ .

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

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

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

Add $2$ and $3$ .

$4 y^{5} (3 x^{2} \sqrt[3]{3 x^{2} y^{2}})$

Multiply $3$ by $4$ .

$12 y^{5} (x^{2} \sqrt[3]{3 x^{2} y^{2}})$

Reorder factors in $12 y^{5} (x^{2} \sqrt[3]{3 x^{2} y^{2}})$ .

$12 y^{5} x^{2} \sqrt[3]{3 x^{2} y^{2}}$