DanA> Am curious to know how "simpler" is defined. --Dan On May 2, 2014, at 6:41 PM, Bill Gosper <billgosper@gmail.com> wrote: . . . a radical denester which simplifies (1 + 3*Sqrt[5] - 2*Sqrt[10 + 2*Sqrt[5]])^(1/4) . . . to √(Sqrt[2] 5^(1/4) - Sqrt[1 + Sqrt[5]]) |. . .] Good question. Corey defines simpler as smaller denominator of outermost power, even if the radicand lengthens. Here, it shortened, so there's no contest, assuming that you got a square root sign in front of the 2nd expression. Rich complains that he and presumably other funsters are getting ?s instead of square root, cube root, Phi, pi, and subscript characters, and quoted my message back so damaged, even though my copy from math-fun looks fine. (Except for !@#$%^&* gratuitous linebreaks, for which I blame GMail.) Anyway, since it was short, here it is again in Old High Cuneiform: [Goaded by Mike Hirschhorn] A couple of years ago, a young friend, Corey Ziegler Hunts, wrote a radical denester which simplifies (1 + 3*Sqrt[5] - 2*Sqrt[10 + 2*Sqrt[5]])^(1/4), the fourth root of a trinomial in your expression for G(e^-(2Pi)), to Sqrt[Sqrt[2] 5^(1/4) - Sqrt[1 + Sqrt[5]]], the square root of a binomial. Similarly for H. Three lines later on p2 are two typos: "surprizingly" and "were were". Corey's younger brother Julian and I wrote a sextic solver which led to (6 Cos[Pi/18] - 6 Cos[Pi/9] + 6 Cos[2Pi/9] - 6 Sin[2Pi/9] - 1)^(1/3) ~ 0.2170953869504416 as the value of Phi_1b(2,2;i/3) on p10. --rwg If you mean how "simpler" is defined to Mathematica, I wonder if anyone knows. It seems to be based on LeafCount and a capricious tree search, influenced, in the case of FullSimplify, by an optionally supplied ComplexityFunction. Simplicity, like Intelligence, is a multidimensional quality that confounds ordering.