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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?

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    $\begingroup$ 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. $\endgroup$ – Jonas Meyer Feb 10 '17 at 21:41
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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.

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  • $\begingroup$ 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? $\endgroup$ – idknuttin Feb 10 '17 at 23:35
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    $\begingroup$ @idknuttin look at the equation before the "i.e." It is just a linear equation in one variable, so I solved it using high school algebra. $\endgroup$ – Adam Hughes Feb 10 '17 at 23:45

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