> What exactly does this mean? If (-1)^(1/4) is meant to be any (or all)
> of the 4 roots of -1, then it seems that 0.7+0.7I is the wrong answer.
> But if it is the right answer, then (-1)^(1/4) really means the
> principal root.
If you also try other examples with Mathematica, it seems
that they choose for (-1)^(1/n), etc the root with the
smallest positive complex argument (and
mathworld.wolfram.com says this is an "informal" definition
of principal root). If you also take for sqrt(i) the common
definition of principal root, then indeed sqrt(i)=
(-1)^(1/4). But, its not clear why they think the later is a
simpler or better form. I guess that most software that
computes powers of complex numbers uses the same convention.
For instance, in gcc cpow gives a principal root in this
sense.
By the way, the documentation for Sqrt[], doesn't mention
what criterion they use to choose the root. I think
sometimes Mathematica errs on the side of oversimplifying to
attract the broadest audience.
John