On Thu, 12 Apr 2007, Jaime E. Villate wrote:
> (-8)^(1/3)=-2 is "wrong" in the same sense that 8^(1/3)=2 is also
> "wrong". If you're thinking in terms of complex numbers, in both cases
> you are choosing just one of the 3 solutions; I do not think
> any of those two statements is wrong, if you are restricted to a real
> domain.
There is a considerable literature about working with multi-valued
functions in CASs, in particular, about simplification. This matter is far
from trivial. In order to use simplification rules, one needs a consistent
choice of cuts and main branches of various functions. See, for example,
papers by Corless "Arccoth need not be uncouth" and other papers.
So, I disagree. 8^(1/3)=2 and (-8)^(1/3)=-2 don't belong to the same sheet
of the Riemann surface. With the "standard" choice of the main branch, the
first equality is correct, and the second one is wrong.
I am sorry, I have not checked the behaviour of maxima; I thought is uses
Corless' conventions, too. MuPAD, for example, gives
>> rectform((-8)^(1/3));
1/2
1 + I 3
and I consider this correct.
Andrey