I was playing around with complex exponentials and came to this result:(adsbygoogle = window.adsbygoogle || []).push({});

[tex]$\begin{eqnarray*}

e^{\frac{2\pi i}{5}}&=&e^\left(\frac{2}{5}\right)\left(\pi i\right)\\

&=&\left(e^{\pi i}\right)^{\frac{2}{5}}\\

&=&\left(-1\right)^{\frac{2}{5}}\\

&=&1\end{eqnarray*}$[/tex]

But obviously [tex]e^{\frac{2\pi i}{5}}=\mathrm{cos}\frac{2\pi}{5}+i \mathrm{sin}\frac{2\pi}{5}\approx 0.309+0.951i\neq 1[/tex]

So after some research I found that the exponential law [tex]a^{mn}=\left(a^{m}\right)^{n}[/tex] is only true when [tex]a,m,n\in\mathbb{R}[/tex] and not otherwise.

My question now is WHY does the index law fail for imaginary base/exponents?

Thanks!

*PS how can I get rid of that [tex](0)[/tex] appearing after the eqnarray in my [tex]$\LaTeX$[/tex] code above?? :P

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# Exponential law and complex numbers

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