# RootReduce and Simplify for an algebraic expression

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?

term = Sqrt[104 Sqrt[6] + 468 Sqrt[10] + 144 Sqrt[15] + 2006];


$$13 \sqrt{2}+4 \sqrt{3}+18 \sqrt{5}$$

Another possible approach

poly = MinimalPolynomial[term, x]
sol = Solve[Factor[poly, Extension -> Sqrt@{2, 3, 5}] == 0, x]
Select[sol, PossibleZeroQ[(x /. #) - term] &]


$$2125355779600 - 13379972960 x^2 + 19011864 x^4 - 8024 x^6 + x^8$$

$$\left\{\left\{x\to -13 \sqrt{2}-4 \sqrt{3}-18 \sqrt{5}\right\},\left\{x\to 13 \sqrt{2}-4 \sqrt{3}-18 \sqrt{5}\right\},\left\{x\to -13 \sqrt{2}+4 \sqrt{3}-18 \sqrt{5}\right\},\left\{x\to 13 \sqrt{2}+4 \sqrt{3}-18 \sqrt{5}\right\},\left\{x\to -13 \sqrt{2}-4 \sqrt{3}+18 \sqrt{5}\right\},\left\{x\to 13 \sqrt{2}-4 \sqrt{3}+18 \sqrt{5}\right\},\left\{x\to -13 \sqrt{2}+4 \sqrt{3}+18 \sqrt{5}\right\},\left\{x\to 13 \sqrt{2}+4 \sqrt{3}+18 \sqrt{5}\right\}\right\}$$

$$\left\{\left\{x\to 13 \sqrt{2}+4 \sqrt{3}+18 \sqrt{5}\right\}\right\}$$

• Thanks so much. I didn't know about RadicalDenest... a great help. ($\checkmark$) Aug 9, 2021 at 2:56
• @DavidG.Stork It's my pleasure. Aug 9, 2021 at 3:13