MathGroup Archive 2001

[Date Index] [Thread Index] [Author Index]

Search the Archive

Re: Problem to evaluate cube root of a negative cube nember where a real value is expected

  • To: mathgroup at smc.vnet.net
  • Subject: [mg28065] Re: [mg28042] Problem to evaluate cube root of a negative cube nember where a real value is expected
  • From: Andrzej Kozlowski <andrzej at platon.c.u-tokyo.ac.jp>
  • Date: Fri, 30 Mar 2001 04:12:18 -0500 (EST)
  • Sender: owner-wri-mathgroup at wolfram.com

Actually what you are doing is not simplifying (-8)^(1/3) at all, because in
Mathematica's notation (-8)^(1/3) is definitely not -2. You can check it as
follows:

In[14]:=
(-8)^(1/3)==-2//Simplify

Out[14]=
False

In fact you can see exactly what (-8)^(1/3) is in terms of radicals:

In[15]:=
RootReduce[(-8)^(1/3)]

Out[15]=
1 + I Sqrt[3]

The point is this. There are three complex third roots of -8. The one
denoted by (-8)^(1/3) is by convention (at least in mathematics a little
beyond High School) taken to be the principal value, which is exactly what
Mathematica does, and it is 1 + I Sqrt[3].

The easiest way to obtain all the roots is to enter them as root objects,
that is in the form Root[#^3+8&,1], Root[#^3+8&,2], Root[#^3+8&,3].
Mathematica orders the roots of a polynomial in such a way that the real
ones come first. Thus in this case we get:

In[16]:=
Table[Root[#^3+8,i],{i,3}]

Out[16]=
{-2, 1 - I Sqrt[3], 1 + I Sqrt[3]}


-- 
Andrzej Kozlowski
Toyama International University
JAPAN

http://platon.c.u-tokyo.ac.jp/andrzej/



on 3/29/01 9:24 AM, Gary at garylga at magix.com.sg wrote:

> Hi,
> 
> Does anyone know a simplier way to simplify (-8)^(1/3)=(-2) other than what
> I did below because no complex answer is expected in the solution.
> 
> In[161]:=
> p=(-8)^(1/3)
> q=Abs[p]
> Level[p,3]
> r=Extract[Level[p,3],2]
> (* if p is real, then p should be as below *)
> q*r
> 
> Out[161]=
> \!\(2\ \((\(-1\))\)\^\(1/3\)\)
> 
> Out[162]=
> 2
> 
> Out[163]=
> \!\({2, \(-1\), 1\/3, \((\(-1\))\)\^\(1/3\)}\)
> 
> Out[164]=
> -1
> 
> Out[165]=
> -2
> 
> 
> ______________________________________
> Gary Lee Guanan (garylga at magix.com.sg)
> Director - Business Development
> IQExplorers.com Pte Ltd
> Tel 874-1345/6
> ========================C/o Address============================
> Incubation Centre, School of Computing(SoC),  NUS.
> S15 #01-09, 1 Science Drive 2 (along Lower Kent Ridge Road), S117543
> ===============================================================
> 
> 
> 



  • Prev by Date: Re: log x > x - proof?
  • Next by Date: Re: Unconventional Max[] behavior
  • Previous by thread: Problem to evaluate cube root of a negative cube nember where a real value is expected
  • Next by thread: RE: Problem to evaluate cube root of a negative cube nember where a real value is expected