All of Maximas and Minimas!

Let $\displaystyle z = re^{i\theta}$

$\left|e\right|$

$\large\displaystyle \begin{aligned} \left| z^i \right| &=\left| r^i \right| \left| e^{-\theta} \right|\\ &=\left| r^i \right| e^{-\theta} \\&=\left| e^{\ln r^i} \right| e^{-\theta}\\&=e^{-\theta} &\left[\because \left|e^{\ln r^i}\right| = \left|e^{i\ln r}\right| = 1\right] \end{aligned}$

Now taking Principle Angles,

$\begin{aligned}-\pi&<\theta\le\pi\\\Rightarrow \pi &> -\theta\ge -pi\\\therefore e^{\pi}&>e^{\theta}\ge e^{-\pi}\end{aligned}$

$\therefore \max + \min = e^{\pi} + e^{-\pi} = \boxed{23.18390655}$