Evaluate, pen and paper only:
sqrt(sqrt(121)-sqrt(120))
121 is the square of an odd number, and 120 is the preceding number. So I'll generalize this into sqrt[sqrt[(2k+1)^2]-sqrt[(2k+1)^2-1]]. Then the specific problem is evaluating this at k=5.
sqrt[sqrt[(2k+1)^2] - sqrt[(2k+1)^2-1]]
= sqrt[(2k+1) - sqrt[4k^2+4k]]
= sqrt[(k+1) - 2*sqrt[k*(k+1)] + k]
= sqrt[sqrt[k+1]^2 - 2*sqrt[k*(k+1)] + sqrt[k]^2]
= sqrt[(sqrt[k+1] - sqrt[k])^2]
= sqrt[k+1] - sqrt[k]
Then at k=5 we have sqrt[sqrt[121] - sqrt[120]] = sqrt[6] - sqrt[5].
Generalized Solution
