Joerg> * Bill Gosper <billgosper@gmail.com <http://gosper.org/webmail/src/compose.php?send_to=billgosper%40gmail.com>> [Feb 11. 2011 09:45]:
[...] Out[250]= (1 - Sqrt[5] + 3 Sqrt[-2 + Sqrt[5]] + (-2 + Sqrt[5])^(3/2))/(2 Sqrt[2])
which canNOT be done by working outward from the innermost.
Your (1 - sqrt(5) + 3*sqrt(-2+sqrt(5)) + (-2+sqrt(5))^(3/2))/(2*sqrt(2)) = 1/2*(sqrt(sqrt(5)-1) - sqrt(3-sqrt(5))) = 0.1188769458... (sqrt of 5th singular value) Yeth, LambdaStar[x]:=Sqrt[ModularLambda[Sqrt[-x]]] is the xth singular value. For the three fifthsth singular value, MathWorld misprints a doubly nested radical containing a 23 that should be 27, which upon correction denests completely to LambdaStar[3/5] = (1 - Sqrt[3] + 3 Sqrt[5] + Sqrt[15])/(8 Sqrt[2]). Bob Baillie>in mathematica 8: c = (1 + Sqrt[3] + Sqrt[2] 3^(3/4))^(1/3)/(Sqrt[3] - 1)^(1/6); FullSimplify[c] = (2*(7 + 3*Sqrt[3] + Sqrt[72 + 42*Sqrt[3]]))^(1/6) which has even fewer radicals than the one given just below. of course, one can always ask, what does "simplest" mean? bob --- For most purposes, nesting depth trumps radical count. Regrettably, FullSimplify just minimizes LeafCount. I haven't tried it in Mma 8, but in 7, tedious penalization of nesting via custom definitions of ComplexityFunction led me to conclude that Mma does not know how to denest. Perhaps there is something infeasible about Landau's 18yr old algorithm. --rwg