how do you calculate the cube root without calculator
There is, I vaguely remember, an algebraic method similar to that for calculating square roots and it's also possible to use the Newton-Raphson method, (though you would need to be good at arithmetic, hardly the sort of calculation you would attempt without a calculator).
If the number is close to some convenient cube, it's possible to make use of the binomial expansion. For example.
That's easily done on paper and is correct to 6dp.
The iterative method Bertie mentioned, to find the cube root of "a", say, is:
This is derived using Newton-Raphson.
However, in general, I wouldn't want to use this without a calculator of some sort!