[math-fun] solving and denesting

Early in the sequence of solvable quintic trinomials comes In[105]:= Squint[5*x^5 + 6*x + 6]; whose only real root is In[107]:= FullSimplify /@ %105[[4]] Out[107]= -(1/5) 2^(3/5) 3^(1/5) - 1/5 2^(1/5) 3^(2/5) - 1/5 2^(4/5) 3^(3/5) + 1/5 2^(2/5) 3^(4/5) Raising this to consecutive powers, we find the third gives a binomial: In[108]:= Table[FullSimplify[%^k], {k, 3}] Out[108]= {Root[6 + 6 #1 + 5 #1^5 &, 1], Root[-36 + 36 #1 + 60 #1^3 + 25 #1^5 &, 1], 1/5 2^(3/5) 3^(1/5) (-3 + 2^(1/5) 3^(2/5))} thus rediscovering one of my favorite denestings: (-2^(1/5) + 3^(3/5))^(1/3) == (2^(2/5) + 3^(1/5) + 2^(3/5) 3^(2/5) - 2^(1/5) 3^(3/5))/5^(2/3) --rwg