The Magnitude of a Real Number Raised to an Imaginary Power Is Always 1
Theorem
$$ \left| r^{i \theta} \right| = 1 $$ For nonzero real numbers $r, \theta$, the magnitude of $r^{\theta}$ raised to an imaginary power is $1$.
Explanation
The fact that $\left| e^{i \theta} \right| = 1$ is commonly well known, but it is hard to recall that the base need not be $e$ in particular—any real number works. If you think about it, it obviously has to hold, but since it is rarely used in this form, it slips the mind; yet it is used as a lemma in far more theorems than you might imagine.
Proof
If $\theta = 0$, then since $r^0=1$, obviously $\left| r^{i \theta} \right| = \left| r^{i \cdot 0} \right| = 1$ holds.
If $\theta \ne 0$, then $$ \left| r^{i \theta} \right| = \left| e^{i \theta \ln r} \right| = \left| e^{i \theta (\text{Log} |r| + i \arg r )} \right| $$ Since $\arg r = 0$, $$ \left| e^{i \theta (\text{Log}|r| + i \arg |r| )} \right| = \left| e^{i \theta \text{Log}|r| } \right| $$ $ \theta^{\prime} := \theta \text{Log}|r|$ is also a real number, and for every real number $\theta '$ we have $\left| e^{i \theta^{\prime}} \right| = 1$, so $$ \left| e^{i \theta \text{Log}|r| } \right| = \left| e^{i \theta^{\prime}} \right| = 1 $$
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