Hello so I know that there are a bunch of videos online on how to solve this but I was messing around in class and got $i = \frac{1}{2}$. Here is what I did:
$i^i=x$,
$i^{-1}=x^2$,
$\frac{1}{i}=x^2$,
$\frac{1}{-1}=x^4$,
$-1=x^4$,
$i=x^2$,
$x=\sqrt(i)=i^\frac{1}{2}$,
$i^i=i^\frac{1}{2}$,
$i=\frac{1}{2}$.
Where did I go wrong?
The third line
$(i^i)^2=i^{2i}$ not $i^{-1}$
Recall $(a^m)^n=a^{mn}$, not $a^\left(m^n\right)$
How I would attack the problem:
$~i~$ can be expressed as $~\exp[ ~i(\pi/2 + 2k\pi) ~] ~: ~k \in \Bbb{Z}.$
Then, $~i^i = \exp\{ ~[ ~i(\pi/2 + 2k\pi) ~] \times i\} = \exp[ ~-(\pi/2 + 2k\pi) ~].$
At this point, you have to examine the context of the problem. For example, a book/class may have already adopted the convention that $~i~$ should be specifically expressed as $~\exp[i\pi/2],~$ which would imply that $~k~$ is (somewhat arbitrarily, by convention) forced to equal $~0,~$ in this situation.
If such a convention is in force, then the problem yields the unique value of $~\exp[ ~-\pi/2~].$
