# Is it possible to take the ith root of some number?

1. Aug 29, 2010

### exclamaforte

And if so, could I take the ith root of i?

2. Aug 29, 2010

### Petr Mugver

$$i^{1/i}=i^{-i}=e^{-i\log i}=e^{-i\log e^{i(\pi/2+2k\pi)}}=e^{-ii(\pi/2+2k\pi)}=e^{(\pi/2+2k\pi)}$$

i has infinitely many ith roots, and they are all real!