Simplify cube root of 64(x+y)^6

$\sqrt[3]{64 {(x + y)}^{6}}$

Rewrite $64 {(x + y)}^{6}$ as ${(4 {(x + y)}^{2})}^{3}$ .

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

Pull terms out from under the radical, assuming real numbers.

$4 {(x + y)}^{2}$

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

$4 ((x + y) (x + y))$

Expand $(x + y) (x + y)$ using the FOIL Method.

Apply the distributive property.

$4 (x (x + y) + y (x + y))$

Apply the distributive property.

$4 (x \cdot x + x y + y (x + y))$

Apply the distributive property.

$4 (x \cdot x + x y + y x + y \cdot y)$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

$4 (x^{2} + x y + y x + y \cdot y)$

Multiply $y$ by $y$ .

$4 (x^{2} + x y + y x + y^{2})$

Add $x y$ and $y x$ .

Reorder $y$ and $x$ .

$4 (x^{2} + x y + x y + y^{2})$

Add $x y$ and $x y$ .

$4 (x^{2} + 2 x y + y^{2})$

Apply the distributive property.

$4 x^{2} + 4 (2 x y) + 4 y^{2}$

Multiply $2$ by $4$ .

$4 x^{2} + 8 x y + 4 y^{2}$