1. The problem statement, all variables and given/known data I've been recapitulating some lessons we learned in high school 2 years ago for the exams I need to take this year. There was this exercise I couldn't solve in a nice way. z^2005+(1/z^2005) if we know that z^2+z+1=0 2. Relevant equations I couldn't came up with a good solution, so I looked at the key at the end of the book, and it said that from z^2+z+1=0 => z^3=1, and then you do z^2005=(z^3)^668+z=1^668+z=1+z 3. The attempt at a solution I calculated the root for the equation, and I found that it is (-1±i√3)/2. Now, that complex number definitely isn't 1 on the third power, and I can't take (1+z)+1/(1+z) too far either. Where did I mess up? Is the key in my book correct? Thank you for your answers.