I just learnt a method for finding cube roots of numbers between 1,000 and 1,000,000 with integer cube roots, and it's pretty easy.
Roughly: Look at the last digit of your number, that determines the second digit of your cube root
Divide by 1,000 and round down, then find the next cube below that number. The cube root of that is the 10's digit.
E.g. 389,017.
A 7 at the end can only be caused by a units digit of 3
389 is larger than 343 (but smaller than 512), which is 7 cubed.
So this is 73^3