Proving $e^{i \theta} =1$

Question

Let $\theta \in (-\pi,\pi) $ we have that $$ e^{i\theta} = \left(e^{i\theta}\right)^\frac{2\pi}{2\pi} = (e^{2\pi i})^{\frac{\theta}{2\pi}} $$ so as $e^{2\pi i}=1$ we have that $$ e^{i\theta} = (1)^{\frac{\theta}{2\pi}} =1 $$ But there must be something wrong because $e^{i\pi}=-1 \neq 1$.