How to rewrite $i^3$ I learned that: $\sqrt[3]i=(e^{\frac{\pi}{2}i+2k\pi i})^\frac{1}{3}=e^{\frac{\pi i}{6}+\frac{2}{3}k\pi i}$ for k={0, 1, 2} 
Now how about this case:
$i^3=(e^{\frac{\pi}{2}i+2k\pi i})^3=e^{\frac{3\pi i}{2}+6k\pi i}$ for k={???}
Why would it be $+6k\pi i$? Why not $+2k\pi i$? It seems that for example $e^{\frac{3\pi i}{2}+2\pi i}$ should also be a solution?
And, for what $k$ is it valid?

When we say  $z = re^{ti + 2k\pi i}$ the $2k\pi i$ is just the period and is always there much as "plus a constant"
So really  $\sqrt[3]i=(e^{\frac{\pi}{2}i+2k\pi i})^\frac{1}{3}=e^{\frac{\pi i}{6}+\frac{2}{3}k\pi i + 2k'\pi i}$  But $\frac{2}{3}k\pi i$ is now significant and means more that simply "give or take a few periods".  But $2k'\pi i$ is redundant.  $e^{\frac{\pi i}{6}+\frac{2}{3}k\pi i + 2k'\pi i}= e^{\frac{\pi i}{6}+\frac{2k+6k'}{3}\pi i }$ so we simply don't need to write the period term.  So we don't.
To go the other way should really be $i^3=(e^{\frac{\pi}{2}i+2k\pi i})^3=e^{\frac{3\pi i}{2}+6k\pi i+ 2j\pi i}$ to get the "give or take some periods" back in.  But that's now excessive.  If we set $j = -2k$ we just get back to $6k\pi i+ 2j\pi i=2k\pi i$.  
tl;dr
If the tacked on $\gamma \pi i$ is such that $\gamma$ is not an integer multiple of $2$, then $\gamma \pi i$  is significant.  If $\gamma$ is  an integer multiple of $2$, then $\gamma \pi i$  is not significant.
