+1776 Squaring the equation z + 1/z = 1 gives z^2 + 2 + 1/z^2 = 1. Then multiplying by z^2 gives z^4 + 2z^2 + 1 = z^2. Rearranging, we get z^4 + 2z^2 + 1 - z^2 = 0, which factors as (z^2 + 1)(z^2 + 1) = 0. Since z^2 + 1 = 0 has no solutions, z^2 = -1. Taking the square root of both sides, we find z = i or z = -i.
If z = i, then z^3 = i^3 = -i.
Alternatively, if z = -i, then z^3 is also equal to -i.
Therefore, in either case, z^3 = -i.
+36 Yeah, I managed to solve it 30 minutes later. (Sorry, the correct answer was -1.)
Anyways, thanks for the answer. What I did was take
z+1z=1 and multiply it by z2, getting z3−z2+z=0. However, when you multiply z+1z=1 by z, you will get
z2−z=1
From which you can replace the −z2+z with −1 (The negative of the equation above) to get
z3−1=0
From which you can easily get z3=−1
