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?

Well,
\begin{align}
a+bi &= re^{i \theta}, \qquad (r=|z|) \\
a-bi &= re^{-i \theta}
\end{align}
Thus. 
\begin{align}
(a+bi)^{n}+(a-bi)^{n} &= \left(re^{i \theta} \right)^{n}+ \left(re^{-i \theta}\right)^{n} \\
&= r^{n}\left( e^{in \theta}+e^{-in \theta}\right) \\
&= 2|z|^{n} \cos n \theta
\end{align}
