Simplify x^3y^3 cube root of 27x^4y^8

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

Rewrite $27 x^{4} y^{8}$ as ${(3 x y^{2})}^{3} (x y^{2})$ .

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

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

Factor out $x^{3}$ .

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

Factor out $y^{6}$ .

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

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

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

Move $x$ .

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

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

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

Add parentheses.

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

Pull terms out from under the radical.

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

Rewrite using the commutative property of multiplication.

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

Multiply $x^{3}$ by $x$ by adding the exponents.

Move $x$ .

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

Multiply $x$ by $x^{3}$ .

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

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

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

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

Add $1$ and $3$ .

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

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

Move $y^{2}$ .

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

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

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

Add $2$ and $3$ .

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