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MathGroup Archive 1997

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Re: Algebra Problem

  • To: mathgroup at smc.vnet.net
  • Subject: [mg7642] Re: [mg7624] Algebra Problem
  • From: seanross at worldnet.att.net
  • Date: Tue, 24 Jun 1997 03:36:05 -0400 (EDT)
  • Sender: owner-wri-mathgroup at wolfram.com

Robert McNally wrote:
> 
> I am using Mathematica to study Algebra. When I solve the following
> equation from my textbook with Mathematica, it gives me this result:
> 
> In[1]:=
> Solve[2x^(2/3) + 3x^(1/3) - 9 == 0]
> 
> Out[1]=
>        27
> {{x -> --}}
>        8
> 
> However, my textbook claims that -27 is also a solution. If I tell
> Mathematica to turn off its solution verification, it finds the other
> textbook solution:
> 
> In[213]:=
> Solve[2x^(2/3) + 3x^(1/3) - 9 == 0, VerifySolutions -> False]
> 
> Out[213]=
>                    27
> {{x -> -27}, {x -> --}}
>                    8
> 
> When I ask Mathematica to substitute -27 for x in the equation,
> Mathematica only goes so far in simplifying the equation, but not far
> enough to determine if the left side is the same as the right side:
> 
> In[2]:=
> 2x^(2/3) + 3x^(1/3) - 9 == 0 /. x -> -27
> 
> Out[2]=
>            1/3          2/3
> -9 + 9 (-1)    + 18 (-1)    == 0
> 
> Now, I can see that the cube root of -1 is -1, and that the taking the
> square of -1 yields 1, and then taking the cube root of that also yields
> 1. So the equation should simplify to:
> 
> -9 - 9 + 18 == 0
> 
> and then
> 
> 0 == 0
> 
> which would indicate that -27 is indeed a solution. But I can't figure out
> how to get Mathematica to return similar results. Trying to force
> Mathematica to give a numerical answer, yields an imaginary, non-zero
> result:
> 
> In[3]:=
> N[2x^(2/3) + 3x^(1/3) - 9 /. x -> -27]
> 
> Out[3]=
> -13.5 + 23.3827 I
> 
> which may explain why Mathematica rejects this solution. So, is
> Mathematica rejecting the solution -27 appropriately or not?
> 
> Thanks in advance.
> 
> ======================================================================
>       Robert McNally <mailto:ironwolfNO at SPAMdangerousgames.com>
>     Visit <http://personalweb.lightside.com/pfiles/mcnally1.html>
> ----------------------------------------------------------------------
>  Finger for my PGP Key -- Protect Your Crypto-Rights! * Free Speech!
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> ======================================================================


Mathematica has no built-in preference over which root to choose when
doing complex operations.  Humans, on the other hand, tend to like real
numbers.  (-27)^(1/3) has three answers as does (-27)^2/3.  Two of those
roots are complex, one is real.  The package 'RealOnly' in the standard
add-on packages may help you out.  Instead of using the Solve command,
you could also check out Roots and FindRoots.


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