# [Maxima] x^(1/3) WAS: strange behaviour with simple decimals

Stavros Macrakis macrakis at alum.mit.edu
Wed Apr 11 15:23:30 CDT 2007

```On 4/11/07, Andrey G. Grozin <A.G.Grozin at inp.nsk.su> wrote:
>
> A classical example is x^(1/3). School children (and even teachers)
> beleive that it is real and negative for x<0. Maxima uses a more
> consistent definition - a cut along a negative real half-axis, with an
> additional rule that when we are exactly on the cut, the value from its
> upper side is used. So, for x<0 the result is complex. I'd say that here
> (as very often) maxima is right, and the school education is wrong.
>

Alas, Maxima isn't consistent.

(-1)^(1/3) => -1
(-1.0)^(1.0/3) => (-1.0)^0.3333
x^(1/3) (with x<0) => sqrt(3)*%i*(-x)^(1/3)/2+(-x)^(1/3)/2

(-1)^(3/4) => (-1)^(3/4) =rectform> %i/sqrt(2)+1/(2*sqrt(2))-3*2^-(3/2)
((-1)^3)^(1/4) => (-1)^(1/4) =rectform> %i/sqrt(2)+1/sqrt(2)
((-1)^(1/4))^3 => (-1)^(3/4)
rectform(x^(3/4)) (x<0) =(x=-1)> %i/sqrt(2)-1/sqrt(2)
rectform(x^(1/4)) (x<0) =(x=-1)> %i/sqrt(2)+1/sqrt(2)

(-1)^(3/2) => -%i
((-1)^(1/2))^3 => -%i
((-1)^3)^(1/2) => %i

I am not sure how one would go about making all this better....

-s
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