Simplify the Radical Expression fourth root of 81x^6y^18

$\sqrt[4]{81 x^{6} y^{18}}$

Rewrite $81 x^{6} y^{18}$ as ${(3 x y^{4})}^{4} (x^{2} y^{2})$ .

Rewrite $81$ as $3^{4}$ .

$\sqrt[4]{3^{4} x^{6} y^{18}}$

Factor out $x^{4}$ .

$\sqrt[4]{3^{4} (x^{4} x^{2}) y^{18}}$

Factor out $y^{16}$ .

$\sqrt[4]{3^{4} (x^{4} x^{2}) (y^{16} y^{2})}$

Rewrite $y^{16}$ as ${(y^{4})}^{4}$ .

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

Move $x^{2}$ .

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

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

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

Add parentheses.

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

Pull terms out from under the radical.

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

Remove non-negative terms from the absolute value.

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

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

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

Rewrite $\sqrt[4]{{(x y)}^{2}}$ as $\sqrt{\sqrt{{(x y)}^{2}}}$ .

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

Pull terms out from under the radical.

$3 y^{4} | x | \sqrt{| x y |}$