a^n*b^n != (a*b)^n
- To: mathgroup@smc.vnet.net
- Subject: [mg12267] a^n*b^n != (a*b)^n
- From: "Michael Milirud" <mmichael@idirect.com>
- Date: Tue, 5 May 1998 03:30:02 -0400
- Organization: "TUCOWS Interactive Inc"
This is not so much about the Mathematica as a software as about mathematica as a subject. Mathematica just confirmed it and I am REALLY puzzled on this one. I always considered it trivial that a^n*b^n == (a*b)^n when a,b are complex and n is real. However: 6 == Sqrt[36] == Sqrt[-4*-9] == Sqrt[-4]*Sqrt[-9] == 2i*3i == 6*i^2 == -6 Hence 6 == -6 ARGHHH!!!! After quite some time, I found the problem to be in the step: Sqrt[-4*-9] == Sqrt[-4]*Sqrt[-9] which as Mathematica claims does NOT equal to each other!!! So generally that would mean: a^n*b^n != (a*b)^n I tried to go and search for the basic proof of this equality. Obviously enough I couldn't find any :( For a, b being real and n being positive integer the equality is obvious. But for other cases - I don't know how to approach it. While playing around with different examples I noticed that the above equality upholds for all the cases except when we have a and b being negative REAL numbers and n being p/q with q=2k ANYTHING at all will be greatly appriciated, as I am completely stuck!!! ;( Michael
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