Values of the complex power $1^\sqrt{2}$

Question

I have to show that the values of the complex power $1^\sqrt{2}$ all lie on the unit circle, i.e. that $|1^\sqrt{2}|=1$.

$1^\sqrt{2} = e ^ {\sqrt{2} \ln{1}}$ by definition, and $\ln{1} = 2k \pi i$ with $ k\in \mathbb{Z}$, so $1^\sqrt{2} = e ^ {2 \sqrt{2} k \pi i} = e ^ {2 \sqrt{2} k} e ^{\pi i} = e^{2 \sqrt{2} k} ( \cos \pi + i \sin \pi ) = e^{2 \sqrt{2} k}$ by Euler's formula, but these complex numbers definitely do not lie on the unit circle. What am I doing wrong?

Answer 1

$e^{ab} \neq e^a e^b$. The latter equals $e^{a+b}$, not $e^{ab}$.