decimal precision
I don't know how precise on the small scale, but I know that knowing pi to the 15th decimal place is enough to accuratly get the circumfrance of the universe to within a thousandth of an inch. So I would say 12 is pretty solid.
There is in fact a way to calculate cube roots by hand. In fact, if you look at the steps closely, you will see that the method is very similar to some methods for calculating square roots by hand. Searching Google will yeild some quick results. Does anyone know how to calculate quartic roots by...