Here's another way to do it (like Gareth's approach), which is at high school level, but sort of requires that you guess part of the solution in advance: (spoiler space) . . . . . . . . . . . Another way is to hypothesize that the expressions (sqrt(108) + 10)^(1/3) and (sqrt(108) - 10)^(1/3) might each be equal to a + b sqrt(3) for some a and b, since sqrt(108) = 6 sqrt(3). Cubing both sides gives +- 10 + 6 sqrt(3) == a (a^2 + 9 b^2) + 3 b (a^2 + b^2) sqrt(3) or equating coefficients: +- 10 == a (a^2 + 9 b^2) 2 == b (a^2 + b^2) I think the problem with general coefficients on the left-hand side is as hard as simplifying the original cubic, but if you guess that a and b are integers, then you quickly get a = 1 and b = 1 for the +10 case and a = -1 and b = 1 for the -10 case, i.e. (sqrt(108) + 10)^(1/3) = 1 + sqrt(3) (sqrt(108) - 10)^(1/3) = -1 + sqrt(3) so the answer is (1 + sqrt(3)) - (-1 + sqrt(3)) = 2. (I can't remember the book I saw this in originally; I think Gareth's approach also seems a bit less brute-force.) --Neil