Simplify cube root of square root of 125x^3y^3

$\sqrt[3]{\sqrt{125 x^{3} y^{3}}}$

Rewrite $125 x^{3} y^{3}$ as ${(5 x y)}^{2} \cdot (5 x y)$ .

Factor $25$ out of $125$ .

$\sqrt[3]{\sqrt{25 (5) x^{3} y^{3}}}$

Rewrite $25$ as $5^{2}$ .

$\sqrt[3]{\sqrt{5^{2} \cdot 5 x^{3} y^{3}}}$

Factor out $x^{2}$ .

$\sqrt[3]{\sqrt{5^{2} \cdot 5 (x^{2} x) y^{3}}}$

Factor out $y^{2}$ .

$\sqrt[3]{\sqrt{5^{2} \cdot 5 (x^{2} x) (y^{2} y)}}$

Move $x$ .

$\sqrt[3]{\sqrt{5^{2} \cdot 5 (x^{2}) y^{2} x y}}$

Move $5$ .

$\sqrt[3]{\sqrt{(5^{2} (x^{2})) y^{2} \cdot 5 x y}}$

Rewrite $(5^{2} (x^{2})) y^{2}$ as ${(5 x y)}^{2}$ .

$\sqrt[3]{\sqrt{{(5 x y)}^{2} \cdot 5 x y}}$

Add parentheses.

$\sqrt[3]{\sqrt{{(5 x y)}^{2} \cdot 5 (x y)}}$

Add parentheses.

$\sqrt[3]{\sqrt{{(5 x y)}^{2} \cdot (5 x y)}}$

Pull terms out from under the radical.

$\sqrt[3]{5 x y \sqrt{5 x y}}$