The American Invitational Mathematics Exam (AIME) 2006 asked contestants to reduce this expression:
$$\sqrt{104 \sqrt{6} + 468 \sqrt{10} + 144 \sqrt{15} + 2006}$$
The answer is $13 \sqrt{2} + 4 \sqrt{3} + 18 \sqrt{5}$, as can be verified by:
Reduce[Sqrt[104 Sqrt[6] + 468 Sqrt[10] + 144 Sqrt[15] + 2006] ==
13 Sqrt[2] + 4 Sqrt[3] + 18 Sqrt[5]]
(* True *)
Alas, the direct approach for Mathematica-based such simplification of the expression in the problem doesn't work:
mySimple = RootReduce[Sqrt[104 Sqrt[6] + 468 Sqrt[10] + 144 Sqrt[15] + 2006]]
and
ToRadicals[mySimple]
I can make progress if I know (or guess) the algebraic extension:
term= Sqrt[104 Sqrt[6] + 468 Sqrt[10] + 144 Sqrt[15] + 2006];
FullSimplify@AlgebraicNumber[term, {Sqrt[2], Sqrt[5], Sqrt[15]}]
But what if I didn't know the extensions?
I've tried a number of methods, based on FullSimplify
, Expand
, and so on, but none work.
Suggestions?