Prove that `(sqrt(3)/(2) +(i)/(2))^(5) + (sqrt(3)/(2) -(i)/(2))^(5)` is purely real.

Explanation

We have `((sqrt(3))/(2) +(i)/(5))^(5) +((sqrt(3))/(2) -(i)/(2))^(5)`
`= (sqrt(3)/(2) + (i)/(2))^(5) + (bar(sqrt(3)/(2) + (i)/(2)))^(5)`
`=(sqrt(3)/(2) + (i)/(5))^(5) + ((bar(sqrt(3)/(2) + (i)/(2)))^(5)) (because (barz)^(n) = bar((z^(n))))`
`= z + barz`, where `z = (sqrt(3)/(2) + (i)/(2))^(5)`
`= 2Re(z)`
Hence, given complex number is real.