Simplify question

*To*: mathgroup at smc.vnet.net*Subject*: [mg71655] Simplify question*From*: "dimitris" <dimmechan at yahoo.com>*Date*: Sun, 26 Nov 2006 03:48:36 -0500 (EST)

The following list of expressions was obtained by following the steps of Tartaglia's solution of the cubic equation with Mathematica. lstcub = {{((-1)^(1/3)*Q)/(R - Sqrt[Q^3 + R^2])^(1/3) + (-1)^(2/3)*(R - Sqrt[Q^3 + R^2])^(1/3), -(Q/(R - Sqrt[Q^3 + R^2])^(1/3)) + (R - Sqrt[Q^3 + R^2])^(1/3), -(((-1)^(2/3)*Q)/(R - Sqrt[Q^3 + R^2])^(1/3)) - (-1)^(1/3)*(R - Sqrt[Q^3 + R^2])^(1/3)}, {-(Q/(R + Sqrt[Q^3 + R^2])^(1/3)) + (R + Sqrt[Q^3 + R^2])^(1/3), ((-1)^(1/3)*Q)/(R + Sqrt[Q^3 + R^2])^(1/3) + (-1)^(2/3)*(R + Sqrt[Q^3 + R^2])^(1/3), -(((-1)^(2/3)*Q)/(R + Sqrt[Q^3 + R^2])^(1/3)) - (-1)^(1/3)*(R + Sqrt[Q^3 + R^2])^(1/3)}}; TableForm[%]//TraditionalForm Although the solution of the reduced cubic equation can be obtained making a tricky observation (see e.g. Leonard E. (Leonard Eugene) Dickson b. 1874. (page 32) , Elementary theory of equations. 1914. available online at http://mathbooks.library.cornell.edu:8085/Dienst?verb=Display&protocol=CGM&ver=1.0&identifier=cul.math/01460001"; target=_blank >http://mathbooks.library.cornell.edu:8085/Dienst?verb=Display&protocol=CGM&ver=1.0&identifier=cul.math/01460001) so what I ask don't play any important role in the solution procedure, I wonder if Mathematica can verify the equalities MapThread[Equal, lstcub] {((-1)^(1/3)*Q)/(R - Sqrt[Q^3 + R^2])^(1/3) + (-1)^(2/3)*(R - Sqrt[Q^3 + R^2])^(1/3) == -(Q/(R + Sqrt[Q^3 + R^2])^(1/3)) + (R + Sqrt[Q^3 + R^2])^(1/3), -(Q/(R - Sqrt[Q^3 + R^2])^(1/3)) + (R - Sqrt[Q^3 + R^2])^(1/3) == ((-1)^(1/3)*Q)/(R + Sqrt[Q^3 + R^2])^(1/3) + (-1)^(2/3)*(R + Sqrt[Q^3 + R^2])^(1/3), -(((-1)^(2/3)*Q)/(R - Sqrt[Q^3 + R^2])^(1/3)) - (-1)^(1/3)*(R - Sqrt[Q^3 + R^2])^(1/3) == -(((-1)^(2/3)*Q)/(R + Sqrt[Q^3 + R^2])^(1/3)) - (-1)^(1/3)*(R + Sqrt[Q^3 + R^2])^(1/3)} which can be justified in view of the results Table[lstcub /. {R -> Random[], Q -> Random[]}, {5}] // Chop Thanks a lot for any response. Dimitris

**Follow-Ups**:**Re: Simplify question***From:*Daniel Lichtblau <danl@wolfram.com>

**Re: Simplify question***From:*Andrzej Kozlowski <akoz@mimuw.edu.pl>