Example: we have the following expansions:
Table[Expand[(1 + Sqrt[2])^k], {k, 1, 6}]
(* {1 + Sqrt[2], 3 + 2 Sqrt[2], 7 + 5 Sqrt[2], 17 + 12 Sqrt[2], 41 + 29 Sqrt[2], 99 + 70 Sqrt[2]} *)
Or, the powers of GoldenRatio:
Table[Expand[((1 + Sqrt[5])/2)^k], {k, 1, 6}]
(* {1/2 + Sqrt[5]/2, 3/2 + Sqrt[5]/2, 2 + Sqrt[5], 7/2 + (3 Sqrt[5])/2, 11/2 + (5 Sqrt[5])/2, 9 + 4 Sqrt[5]} *)
My questions are:
How to convert an expression 99 + 70 Sqrt[2] back to (1 + Sqrt[2])^6 ?
How to convert (11 + 5*Sqrt[5])/2 back to GoldenRatio^5 ?
In general, how to factor a + b*Sqrt[c] to a power format (if it exists) ?
UPDATE: Thanks to all for the inspiration!
My problem is to solve
a + b*Sqrt[c] == (r + s*Sqrt[c])^k
where r, s and k are unknown and all variables are integer.
Unfortunately I get from
Solve[(99 + 70 Sqrt[2]) == Expand[(r + s*Sqrt[2])^k] && k > 0, {r, s, k}, Integers]
an error message: "This system cannot be solved with the methods available to Solve."
Of course, brute force method over integers is possible, but has anybody an idea, how to solve this equation in integers, for example some additional options in the "Solve[]" statement ?
However, brute force is not good method for simplification of
11880235445948416 + 2206039924408320*Sqrt[29]