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Re: Question
*To*: mathgroup at smc.vnet.net
*Subject*: [mg25266] Re: [mg25258] Question
*From*: Andrzej Kozlowski <andrzej at tuins.ac.jp>
*Date*: Sun, 17 Sep 2000 17:33:35 -0400 (EDT)
*Sender*: owner-wri-mathgroup at wolfram.com
Actually, I forgot to mention that this problem is almost trivial, provided
you do not insist on Mathematica doing all the work. We of course assume
that m is a rational number. This is how it goes:
In[1]:=
Factor[x^3 + 8]
Out[1]=
2
(2 + x) (4 - 2 x + x )
In[2]:=
% /. x -> m^(1/3)
Out[2]=
1/3 1/3 2/3
(2 + m ) (4 - 2 m + m )
So the answer will be
Out[25]=
1/3 2/3
4 - 2 m + m
-----------------
8 + m
Mathematica can also verify this
In[27]:=
FullSimplify[(4 - 2*m^(1/3) + m^(2/3))/(m + 8) ==
1/(m^(1/3) + 2), Element[m, Reals]]
Out[27]=
True
Now I am getting worried if I have not inadvertently done your homework for
you ...
on 00.9.17 6:17 PM, Andrzej Kozlowski at andrzej at tuins.ac.jp wrote:
> on 00.9.17 5:47 PM, Steven Spear at smitsky at mindspring.com wrote:
>
>> Hi, can anyone be so kind as to tell me how this problem:
>>
>> 1/cbrt(m) +2 (One over the Cube Root of "m" plus Two)
>>
>> ...can be solved in Mathematica by rationalizing the denominator? Thank you.
>> Steve
>>
>>
>>
>>
>
> As long as your m is a number the following will work for roots of degree less
> than 5:
>
> RationalizeDenominator1[expr_] :=
>
> FullSimplify[expr, ComplexityFunction ->
> (
> Count[#, _?
> (MatchQ[Denominator[#], Power[_, _Rational] _. + _.] &),
> {0, Infinity}
> ] + If[FreeQ[#, Root], 0, 1] &
> )
> ]
>
> For example:
>
> In[3]:=
> RationalizeDenominator1[1/(3^(1/3) + 2)]
>
> Out[3]=
> 1 1/3 2/3
> -- (4 - 2 3 + 3 )
> 11
>
> or
>
> In[4]:=
> RationalizeDenominator1[1/(7^(1/4) - 3)]
>
> Out[4]=
> 1
> -- (-27 - 3 Sqrt[7] - Sqrt[2 (63 + 44 Sqrt[7])])
> 74
>
>
> It can also be one for expressions involving roots of degree higher than 5 but
> it is considerably more complicated. In fact already once explained how to do
> this in this list (in 1999).
--
Andrzej Kozlowski
Toyama International University
JAPAN
http://platon.c.u-tokyo.ac.jp/andrzej/
http://sigma.tuins.ac.jp/
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