Simplify cube root of 2xy^2+ cube root of 16xy^5+ cube root of 2x^4y^2

$\sqrt[3]{2 x y^{2}} + \sqrt[3]{16 x y^{5}} + \sqrt[3]{2 x^{4} y^{2}}$

Step 1

Rewrite $16 x y^{5}$ as ${(2 y)}^{3} \cdot (2 x y^{2})$ .

Factor $8$ out of $16$ .

$\sqrt[3]{2 x y^{2}} + \sqrt[3]{8 (2) x y^{5}} + \sqrt[3]{2 x^{4} y^{2}}$

Rewrite $8$ as $2^{3}$ .

$\sqrt[3]{2 x y^{2}} + \sqrt[3]{2^{3} \cdot 2 x y^{5}} + \sqrt[3]{2 x^{4} y^{2}}$

Factor out $y^{3}$ .

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

Move $x$ .

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

Move $2$ .

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

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

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

Add parentheses.

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

Add parentheses.

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

Step 2

Pull terms out from under the radical.

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

Step 3

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

Factor out $x^{3}$ .

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

Reorder $2$ and $x^{3}$ .

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

Add parentheses.

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

Add parentheses.

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

Step 4

Pull terms out from under the radical.

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