# Solve the equation $(z+1)^5=z^5$ [duplicate]

Solve the equation $(z+1)^5=z^5$
Since we are dealing with complex numbers I don't think we can take the fifth root of both sides. I can change the right side of the equation to $r^5e^{5i\theta}$, but I don't know what to do with the left side. I don't think substituting $z$ with $x+yi$ would help me, besides foiling out the left side I don't know what to do. Any ideas?

• You can take absolute values and then take positive fifth roots to get a big restriction: You'll know the real part, and from there it simplifies. – Jonas Meyer Feb 10 '17 at 21:41

You can take fifth roots, you just have to note that $z^5=w^5\iff z= e^{2\pi ik/5}w=\zeta_5^k w$ for some $0\le k\le 4$.
So $\zeta_5^kz = z+1$ i.e.
$$z= {1\over \zeta_5^k-1},\quad 1\le k\le 4$$
as clearly $z+1=z$ is impossible.
• Just one question, how do you know $z=\frac{1}{\zeta^k_5 -1}$? Is this a definition so it's always the case or specific to this problem? – idknuttin Feb 10 '17 at 23:35