Simplify 6y^2 cube root of 189x^4y^3

$6 y^{2} \sqrt[3]{189 x^{4} y^{3}}$

Rewrite $189 x^{4} y^{3}$ as ${(3 x y)}^{3} \cdot (7 x)$ .

Factor $27$ out of $189$ .

$6 y^{2} \sqrt[3]{27 (7) x^{4} y^{3}}$

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

$6 y^{2} \sqrt[3]{3^{3} \cdot 7 x^{4} y^{3}}$

Factor out $x^{3}$ .

$6 y^{2} \sqrt[3]{3^{3} \cdot 7 (x^{3} x) y^{3}}$

Move $x$ .

$6 y^{2} \sqrt[3]{3^{3} \cdot 7 (x^{3}) y^{3} x}$

Move $7$ .

$6 y^{2} \sqrt[3]{(3^{3} (x^{3})) y^{3} \cdot 7 x}$

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

$6 y^{2} \sqrt[3]{{(3 x y)}^{3} \cdot 7 x}$

Add parentheses.

$6 y^{2} \sqrt[3]{{(3 x y)}^{3} \cdot (7 x)}$

Pull terms out from under the radical.

$6 y^{2} (3 x y \sqrt[3]{7 x})$

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

Move $y$ .

$6 (y \cdot y^{2}) (3 x \sqrt[3]{7 x})$

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

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

$6 (y^{1} y^{2}) (3 x \sqrt[3]{7 x})$

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

$6 y^{1 + 2} (3 x \sqrt[3]{7 x})$

Add $1$ and $2$ .

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

Multiply $3$ by $6$ .

$18 y^{3} x \sqrt[3]{7 x}$