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From: Miguel on 24 Dec 2008 06:02
Let the equation x^(2/3)-x^(1/3)-6=0

The roots of this equation are x=27 and x=-8. But Mathematica 6.0.1
yields:

In[]: Solve[x^(2/3)-x^(1/3)-6=0,x]
Out[]: {{x->27}}

In[]: x^(2/3)-x^(1/3)-6/.x->27
Out[]: 0

In[]:= x^(2/3)-x^(1/3)-6/.x->-8.
Out[]: -9+1.73205i

Where is my error?


From: Jean-Marc Gulliet on 25 Dec 2008 03:54
Miguel wrote:

> Let the equation x^(2/3)-x^(1/3)-6=0
>
> The roots of this equation are x=27 and x=-8. But Mathematica 6.0.1
> yields:
>
> In[]: Solve[x^(2/3)-x^(1/3)-6=0,x]
> Out[]: {{x->27}}
>
> In[]: x^(2/3)-x^(1/3)-6/.x->27
> Out[]: 0
>
> In[]:= x^(2/3)-x^(1/3)-6/.x->-8.
> Out[]: -9+1.73205i
>
> Where is my error?

It looks like the equation you are solving has only one root. Indeed the
expression x^(2/3) - x^(1/3) - 6 is real only for x in [0, +inf). Note
that Solve, Reduce, and FindRoot, return only one root at 27 (see
examples below).

In[1]:= expr = x^(2/3) - x^(1/3) - 6;
Solve[expr == 0, x]
expr /. x -> 27
expr /. x -> -8 // Simplify

Out[2]= {{x -> 27}}

Out[3]= 0

Out[4]= -9 + I Sqrt[3]

In[5]:= Reduce[expr == 0, x, Complexes]

Out[5]= x == 27

In[6]:= Reduce[expr > 0, x, Reals]

Out[6]= x > 27

In[7]:= Reduce[expr < 0, x, Reals]

Out[7]= 0 <= x < 27

In[8]:= FindRoot[expr == 0, {x, -5}]
% // Chop

Out[8]= {x -> 27.+ 3.15544*10^-30 I}

Out[9]= {x -> 27.}

In[10]:= FindRoot[expr == 0, {x, -30}]
% // Chop

Out[10]= {x -> 27.- 3.27156*10^-26 I}

Out[11]= {x -> 27.}

In[12]:= Plot[expr, {x, -30, 30}]

In[13]:= $Version

Out[13]= "6.0 for Mac OS X x86 (64-bit) (May 21, 2008)"

Regards,
-- Jean-Marc

From: dimitris on 25 Dec 2008 03:55
On 24 =C4=E5=EA, 13:02, Miguel <misv...(a)gmail.com> wrote:
> Let the equation x^(2/3)-x^(1/3)-6=0
>
> The roots of this equation are x=27 and x=-8. But Mathematica 6.0.1
> yields:
>
> In[]: Solve[x^(2/3)-x^(1/3)-6=0,x]
> Out[]: {{x->27}}
>
> In[]: x^(2/3)-x^(1/3)-6/.x->27
> Out[]: 0
>
> In[]:= x^(2/3)-x^(1/3)-6/.x->-8.
> Out[]: -9+1.73205i
>
> Where is my error?

No bug here.

In[12]:=
x^(2/3) - x^(1/3) - 6 /. x -> -8
ComplexExpand /@ {(-1)^(1/3), (-1)^(2/3)}

Out[12]=
-6 - 2*(-1)^(1/3) + 4*(-1)^(2/3)

Out[13]=
{1/2 + (I*Sqrt[3])/2, -(1/2) + (I*Sqrt[3])/2}

Dimitris

From: sjoerd.c.devries on 25 Dec 2008 03:55
The answer can be found in this thread:
http://www.google.com/url?url=http://groups.google.com/g/3e57ff08/t/5e5cd=
1f83420b520/d/ac7edc59dafdb691%3Fq%3D%23ac7edc59dafdb691&ei=k2dSSb65PIysQ=
Lv1idED&sa=t&ct=res&cd=2&source=groups&usg=AFQjCNEpLUjvGSQrLk0RHT=
XChU-klRdSnw

Cheers -- Sjoerd

On Dec 24, 1:02 pm, Miguel <misv...(a)gmail.com> wrote:
> Let the equation x^(2/3)-x^(1/3)-6=0
>
> The roots of this equation are x=27 and x=-8. But Mathematica 6.0.1
> yields:
>
> In[]: Solve[x^(2/3)-x^(1/3)-6=0,x]
> Out[]: {{x->27}}
>
> In[]: x^(2/3)-x^(1/3)-6/.x->27
> Out[]: 0
>
> In[]:= x^(2/3)-x^(1/3)-6/.x->-8.
> Out[]: -9+1.73205i
>
> Where is my error?


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