The Magnitude of the Imaginary Power of a Real Number is Always 1
Theorem
$$ \left| r^{i \theta} \right| = 1 $$ For any real number $r, \theta$ that is not $0$, the magnitude of its imaginary power is $1$.
$$ \left| r^{i \theta} \right| = 1 $$ For any real number $r, \theta$ that is not $0$, the magnitude of its imaginary power is $1$.