Simplifying sum of powers of conjugate pairs The result of summing a conjugate pair of numbers each raised to the power $n$:
$$
(a + bi)^n + (a - bi)^n
$$
Produces a real number where $a + bi$ is a complex number.
Given the result is real, is there a simplified way to express the above expression in terms of $a$ and $b$ involving no imaginary number $i$ in the simplified expression?

Let $z=|z|(cos (\alpha) + isin(\alpha))$ therefore $z^n=|z|^n(cos (n \alpha) + isin (n\alpha))$ so the expression becomes $2|z|^ncos (n\alpha)$. We know $|z|=\sqrt {a^2+b^2}$, also $\alpha$ can be expressed in terms of $a,b$ (using cotangent).
