Simplify cube root of square root of 243x^12

$\sqrt[3]{\sqrt{243 x^{12}}}$

Rewrite $243 x^{12}$ as ${(9 x^{6})}^{2} \cdot 3$ .

Factor $81$ out of $243$ .

$\sqrt[3]{\sqrt{81 (3) x^{12}}}$

Rewrite $81$ as $9^{2}$ .

$\sqrt[3]{\sqrt{9^{2} \cdot 3 x^{12}}}$

Rewrite $x^{12}$ as ${(x^{6})}^{2}$ .

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

Move $3$ .

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

Rewrite $9^{2} {(x^{6})}^{2}$ as ${(9 x^{6})}^{2}$ .

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

Pull terms out from under the radical.

$\sqrt[3]{9 x^{6} \sqrt{3}}$

Rewrite $9 x^{6} \sqrt{3}$ as ${(x^{2})}^{3} \cdot (9 \sqrt{3})$ .

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

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

Reorder $9$ and ${(x^{2})}^{3}$ .

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

Add parentheses.

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

Pull terms out from under the radical.

$x^{2} \sqrt[3]{9 \sqrt{3}}$