Solving the equation $z^7=-1$ Solve the equation $z^7=-1$


My attempt:

$$z=x+yi$$

$$(x+yi)^7+1=0$$

$$(x^2+2yi-y^2)^3(x+yi)+1=0$$

but now it's start to look ugly.

I'm sure that there is a simple way
We can say that $$z^7=-1$$
or,$$z^7=\cos\pi+i\sin\pi$$
or,$$z^7=\cos(4n+2)\pi+i\sin(4n+2)\pi \,\ \text{where} \,\ n=0,1,2,3,4,5,6$$ 
or,$$z=\left[\cos(4n+2)\pi+i\sin(4n+2)\pi\right]^{\frac{1}{7}} \,\ \text{where} \,\ n=0,1,2,3,4,5,6$$
or,$$z=\cos(\frac{4n+2}{7})\pi+i\sin(\frac{4n+2}{7})\pi \,\ \text{where} \,\ n=0,1,2,3,4,5,6$$
HINT:
Try using this,(one of the solution)

$$
z^7=e^{i\pi}
\\z=e^{i\frac{\pi}{7}}
$$