Simplify 6y cube root of 54x^10y^6

$6 y \sqrt[3]{54 x^{10} y^{6}}$

Rewrite $54 x^{10} y^{6}$ as ${(3 x^{3} y^{2})}^{3} \cdot (2 x)$ .

Factor $27$ out of $54$ .

$6 y \sqrt[3]{27 (2) x^{10} y^{6}}$

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

$6 y \sqrt[3]{3^{3} \cdot 2 x^{10} y^{6}}$

Factor out $x^{9}$ .

$6 y \sqrt[3]{3^{3} \cdot 2 (x^{9} x) y^{6}}$

Rewrite $x^{9}$ as ${(x^{3})}^{3}$ .

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

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

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

Move $x$ .

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

Move $2$ .

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

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

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

Add parentheses.

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

Pull terms out from under the radical.

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

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

Move $y^{2}$ .

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

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

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

$6 (y^{2} y^{1}) (3 x^{3} \sqrt[3]{2 x})$

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

$6 y^{2 + 1} (3 x^{3} \sqrt[3]{2 x})$

Add $2$ and $1$ .

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

Multiply $3$ by $6$ .

$18 y^{3} x^{3} \sqrt[3]{2 x}$