There is a lot of sites online showing you how to find cube root of a perfect cube, provided the cubic root is less than 1000. This method requires the memorization of 10 cubes from 1 to 10. This is easy.
The problem arises when we are really talking about finding the cube root of all real numbers. Which a lot of those websites do not claim but erroneously titled as such.
So, this project will look into this matter. It would seem that Shakuntala Devi managed to find the roots of larger perfect cubes, but the cubes must still be perfect to begin with.
Basic principle: (x+n)^3 = a, given a, find (x+n)
Attempt 1:
Let x be "left most digit" times 10^{\text{number of digits in n}} + n, where n is the rest of the digits.
Stuck at:
3x^2n + 3xn^2 + n^3 = a - x^3
RHS is easy to obtain, using the perfect cube root trick (memorize cubes from 1 to 10).
However, we need to find n, which is virtually impossible to isolate. Although, for any real, I could figure out the first two leftmost digits of the solution: \frac{a-x^3}{3xn(x+n)}. First do long division with 3x, then division by n(x+n) where n is easily found, by brute force for no more than 10 times.
Attempt 2:
let x = n, hence we need to find 2x.
To be continued...