Simplifying expressions such as $e^{-62\pi i/7}$ and $e^{2000\pi i/15}$. Dealing with multiples of $2\pi$ in the argument.

Question

What are the specific steps or rules you should follow when simplifying the argument of a complex number? I am having trouble figuring out the exact methodology when dealing with multiples of $2\pi$ in particular.

Example problems: $$e^{-62\pi i/7}$$ $$e^{2000\pi i/15}$$

I have not found any good resources with worked out solutions in general for arguments that are multiples of $2\pi$.

Answer 1

Keep in mind that the exponential function is periodic with period $2\pi i$. So: \begin{align} \exp\left(-\frac{62\pi i}7\right)&=\exp\left(-\frac{70\pi i}7+\frac{8\pi i}7\right)\\ &=\exp\left(-10\pi i+\frac{8\pi i}7\right)\\ &=\exp\left(\frac{8\pi i}7\right) \end{align} and \begin{align} \exp\left(\frac{2000\pi i}{15}\right)&=\exp\left(\frac{400\pi i}3\right)\\ &=\exp\left(\frac{396\pi i}3+\frac{4\pi i}3\right)\\ &=\exp\left(132\pi i+\frac{4\pi i}3\right)\\ &=\exp\left(\frac{4\pi i}3\right). \end{align}