I do not know any tricks to actually solve this, but you can get a fairly good estimate using basic calculus. In particular, note that 31^{2} < (30*31*32*33 + 1)^{1/2} < 32^{2}. We can model √ around 31^{4} linearly by the function defined by L(x) = 31^{2} + (2*31^{2})^{1}(x31^{4}). Take x = 31.5^{4} as an initial guess and we get something like L(x) ≈ 992. I think that actual answer is 991, so this gets you a pretty good estimate and all of the calculations are manageable.
I am sure someone else knows a nice method to tackle this problem though, so I would like to see that as well.
