This is something that's been bothering me for some time, and I thought starting a post on in might help me out. I'm not posting it in the "Help Me" forum, because instead of a question, I was hoping for an open dialog. If some one feels this is the wrong place, then pleas move it to "Help Me."

This is as far as I've gotten:

From that:

That said:

Also:

Which shows:

With a similar method (that is kind of tedious to do out) I found:

I get how to find the kth root of any imaginary number, but I can't make a general rule, I have to do each out by hand.

Using Taylor series I can show :

And from there I think I get k^i, but only if k is a real number.

which leaves me with the following questions:

That last one is a calculator answer, but why?

Any other I properties would be fun to see, so if anyone else has something to offer, please, I'd like to learn as much as I can about complex numbers.