Problem: Is it true that $\left\vert a^b \right\vert = \left\vert a \right\vert ^{\vert b \vert }$ for every complex number $a$ and $b$?
Solution: No this need not be true. For example, \[ \left\vert e^\iota \right\vert = 1, \] but \[ \left\vert e \right\vert ^{\vert \iota \vert } = \left\vert e \right\vert ^1 = e. \]