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Re: Mathematica strange behaviour finding a cubic root

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• Subject: [mg129134] Re: Mathematica strange behaviour finding a cubic root
• From: Andrzej Kozlowski <akozlowski at gmail.com>
• Date: Mon, 17 Dec 2012 02:58:51 -0500 (EST)
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You can't because they are not equal of course. Fractional powers are 
defined as

x^a = Exp[a*Log[x]

where Log is the principal branch of the logarithm. It is impossible to 
define a continuous branch of the logarithm in the entire complex plane 
so as you go around it there has to be a "jump" somewhere. The usual 
choice of the so called principal branch makes the jump take place on 
the negative real axis. The two answers that you get to yoru computation 
come from different branches of the logarithm. In fact here is one of 
your answers:

 Exp[(1/3)*Log[1/4]]

 1/2^(2/3)

and here is the other:

Simplify[Exp[(1/3)*(Log[1/4] + 2*Pi*I)]]

 (-(1/2))^(2/3)

they are certainly not equal. The reason why you think they are equal is 
because you are assuming that

(x^a)^b = x^(a b)

but this is not always true. In fact Mathematica itself can find examples when this does not hold, e.g:


FindInstance[x^(a*b) != (x^a)^b && Element[{x, b}, Reals] && Element[a, Integers], {x, a, b}]

{{x -> -(109/5), a -> 22, b -> -(56/5)}}

Andrzej Kozlowski


On 16 Dec 2012, at 07:06, sergio_r at mail.com wrote:

>
> How can I make Mathematica provides the same answer for
> (-1/2)^(2/3) = ((-1/2)^2)^(1/3) ?
>
> What follows is a Mathematica session:
>
> In[1]:= (-1/2)^(2/3)
>
>           1  2/3
> Out[1]= (-(-))
>           2
>
> In[2]:= N[%]
>
> Out[2]= -0.31498 + 0.545562 I
>
> In[3]:= ((-1/2)^2)^(1/3)
>
>         -(2/3)
> Out[3]= 2
>
> In[4]:= N[%]
>
> Out[4]= 0.629961
>
>
> Sergio
>

• Follow-Ups:
  • Re: Mathematica strange behaviour finding a cubic root
    • From: Murray Eisenberg <murray@math.umass.edu>

• References:
  • Mathematica strange behaviour finding a cubic root
    • From: sergio_r@mail.com