# Relation between $|z^x|$ and $|z|^x$

In the answers given to this question, the following relation is often used:

$$\left| z^x \right| = \left| z \right|^x$$

with $z \in \mathbb{C}$, $z = \alpha + i \beta$.

• How to prove it?
• Can $x$ be a complex number itself, or should $x$ only be a rational number?
• $i^i \neq 1^i$ so $x$ has to be real – Archis Welankar Feb 1 '16 at 8:48
• @ArchisWelankar please, read the comment I wrote in the corindo answer. – BowPark Feb 1 '16 at 9:17
• Someting seems to be wrong – Archis Welankar Feb 1 '16 at 9:19
• @ArchisWelankar tried to fix it. – BowPark Feb 1 '16 at 9:22

You can write $z$ in the form $z = re^{i\theta}$ with $r \in \mathbb{R}_+, \theta \in \mathbb{R}$.
$$\forall x \in\mathbb{R}, \,\, |z^x| = r^x = |z|^x.$$
As noticed @Archis Welankar, when $x$ is complex the relation is not always true.
• Thank you. But for $x \in \mathbb{C}$ I can't see it. If $x = u + iv$, $$(z)^x = (z)^{u + iv} = (r e^{i \theta})^{u + iv} = r^u r^{iv} e^{i \theta u} e^{i \theta iv} = r^u r^{iv} e^{i \theta u} e^{- \theta v}$$ and...? – BowPark Feb 1 '16 at 9:21