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 312 < (30*31*32*33 + 1)1/2 < 322. We can model √ around 314 linearly by the function defined by L(x) = 312 + (2*312)-1(x-314). Take x = 31.54 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.