Simplify ( cube root of x^2y^3z^4)^2

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

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

Factor out $z^{3}$ .

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

Move $x^{2}$ .

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

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

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

Add parentheses.

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

Pull terms out from under the radical.

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

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

Apply the product rule to $y z \sqrt[3]{x^{2} z}$ .

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

Apply the product rule to $y z$ .

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

Rewrite ${\sqrt[3]{x^{2} z}}^{2}$ as ${({(x^{2} z)}^{2})}^{\frac{1}{3}}$ .

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

Apply the product rule to $x^{2} z$ .

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

Multiply the exponents in ${(x^{2})}^{2}$ .

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

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

Multiply $2$ by $2$ .

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

Rewrite $x^{4} z^{2}$ as $x^{3} (x z^{2})$ .

Factor out $x^{3}$ .

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

Add parentheses.

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

Pull terms out from under the radical.

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