This problem was from the 2004 Luzerne County Math Contest:
If (r + 1/r)^2 = 3, what is r + (1/r^3) = ?
Any help would really be appreciated. Thanks in advance!
You can solve
which has complex roots but which can be represented in exponential form quite nicely.
and simplify, and if what I have done has no algebraic errors you should end up with
(note: in a competition you will get no marks for the final answer but just from the method you used to find it)