How do I solve a problem with term $a^{n} + b^{n}$? 
Given two non-zero numbers $x$ and $y$ such that $x^{2} + xy + y^{2} = 0$.
Find the value of
$$\left(\frac{x}{x + y}\right)^{2013} + \left(\frac{y}{x + y}\right)^{2013}$$.

I found out that $(x + y)^2 = xy$ and I'm stuck at $\frac{x^{2013} + y^{2013}}{(x + y)^{2013}}$
Does anyone know how to solve this?

We have 
$${y}^{2}=-{x}^{2}-xy\\
{y}^{3}=-{x}^{2}y-xy^2
$$
Then after substition and elimination $y, y^2,y^3$ we get
$$
(x+y)^3={x}^{3}+3\,{x}^{2}y+3\,x{y}^{2}+{y}^{3}=-x^3.
$$
Thus
$$
\left(\dfrac x{x+y}\right)^{2013}=\left(\left(\dfrac x{x+y}\right)^{3}\right)^{671}=\left(\dfrac{ x^3}{(x+y)^3}\right)^{671}=\left(\dfrac{ x^3}{-x^3}\right)^{671}=-1.
$$
Similarly $$\left(\dfrac y{x+y}\right)^{2013}=-1.$$
