@Ian Fowler my question was actually x^1024+x^-2014 not x^1024+x^-1024 but good solution tho
So right you are. Thanks for the correction. Sometimes I'm in too much of a hurry. The same method can be used.
x + 1/x = sqrt(3)
Multiply by x and use the Quadratic Formula. x is not real.
x = sqrt(3)/2 + i/2 or x = sqrt(3) - i/2
I'll use the first one.
in polar form x = sqrt(3) /2 + i/2 = 1*cis(pi/6)
Therefore by DeMoivre: x^1024 = 1*cis(1024pi/6)
= -1/2 + isqrt(3)/2
= cis(-2pi/3) =
-1/2 - i sqrt(3)
And we get x^1024 + x^-1024 = (-1/2 +isqrt(3)/2) + (-1/2 - isqrt(3)/2) = -1
I'll leave x = sqrt(3)/2 - i/2 for you. You should get the same thing.