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MathGroup Archive 1997

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Re: Algebra Problem

  • To: mathgroup at smc.vnet.net
  • Subject: [mg7676] Re: [mg7624] Algebra Problem
  • From: Allan Hayes <hay at haystack.demon.co.uk>
  • Date: Thu, 26 Jun 1997 01:36:59 -0400 (EDT)
  • Sender: owner-wri-mathgroup at wolfram.com

>From: ironwolfNO at SPAMdangerousgames.com (Robert McNally)
>Subject: [mg7676] [mg7624] Algebra Problem

Robert,

For Mma, (-1)(1/3) is

ComplexExpand[(-1)^(1/3)]
	1/2 + (I Sqrt[3])/2
	
r2 =ComplexExpand[(-1)^(2/3)]
-(1/2) + (I Sqrt[3])/2
Which gives
           1/3                2/3
-9 + 9 (-1)    + 18 (-1)

as
-9+9r1 +18 r2//Expand

	-(27/2) + (27 I Sqrt[3])/2
	
You can use an equivelent pair of equations - inlude the definition  
of the cube roots:

Solve[{2u^2 + 3u - 9 == 0, u^3==x},x]

	{{x -> -27}, {x -> 27/8}}

	
Allan Hayes
hay at haystack.demon.co.uk
http://www.haystack.demon.co.uk/training.html
voice:+44 (0)116 2714198
fax: +44 (0)116 2718642
Leicester,  UK


****

Begin forwarded message:

>From: ironwolfNO at SPAMdangerousgames.com (Robert McNally)
>Subject: [mg7676] [mg7624] Algebra Problem
>Organization: Somewhere Out There


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>
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