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/