I'm having a little trouble with this problem -
Find all z such that $z^7=(z+1)^7$.
I've moved $(z+1)$ to the other side to obtain $(\frac {z}{z+1})^7=1,$ but I'm not sure how to carry on. I've tried converting to complex exponential form, but how would I simplify and solve? Thanks!
$$z^7=(z+1)^7$$ $$1=\left(\frac{z+1}{z}\right)^7$$ $$1=\left(1+\frac{1}{z}\right)^7$$
$z=0$ is not the solution of the equation because we obtain in the first equation
$0=1$
Thus $$1+\frac1z=e^{\frac{2\pi i n}{7}},\ \ n=1,2,3,4,5,6$$
$$z=\frac1{e^{\frac{2\pi i n}{7}}-1},\ \ n=1,2,3,4,5,6$$
Here $n\ne0$ because when we put $n=0$ in our solution we get $1+1/z=1$ thus $1/z=0$