Re: Mathematica strange behaviour finding a cubic root

*To*: mathgroup at smc.vnet.net*Subject*: [mg129157] Re: Mathematica strange behaviour finding a cubic root*From*: Murray Eisenberg <murray at math.umass.edu>*Date*: Tue, 18 Dec 2012 02:41:24 -0500 (EST)*Delivered-to*: l-mathgroup@mail-archive0.wolfram.com*Delivered-to*: l-mathgroup@wolfram.com*Delivered-to*: mathgroup-newout@smc.vnet.net*Delivered-to*: mathgroup-newsend@smc.vnet.net*References*: <20121216060645.D7C256924@smc.vnet.net> <20121217075851.C00B8694D@smc.vnet.net>

Which raises the possibly interesting question: how did Mathematica obtain the result for that FindInstance -- with what would seem unexpectedly large integers in the fractions? On Dec 17, 2012, at 2:58 AM, Andrzej Kozlowski <akozlowski at gmail.com> wrote: > 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 >> > --- Murray Eisenberg murray at math.umass.edu Mathematics & Statistics Dept. Lederle Graduate Research Tower phone 413 549-1020 (H) University of Massachusetts 413 545-2838 (W) 710 North Pleasant Street fax 413 545-1801 Amherst, MA 01003-9305

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

**Re: Mathematica strange behaviour finding a cubic root***From:*Andrzej Kozlowski <akozlowski@gmail.com>