Evaluate perfect $\sqrt[3]{X}$ 
Is there a method of working out the perfect cube root of a 3 digits number?

Working out the perfect cube root of a 2 digits number which I know.
An example of $$\sqrt[3]{12167}$$


*

*Cross out $16$ (always two digits before the last digit)

*take the cube root of $12$ is bewtween $2$ and $3$, so it is $2$

*$27$ or $23$

*$23$ because $3^3=27$

Solving equations is equivalent to root finding (as in points where a function equals, not the kind of root in the question).
For any monotonic function a good way to find an integer root if you know the number of digits is to progressively start at the maximal digit and increase one by one until you find it changes sign or equals 0. The digit just before it changes sign is that nth digit. Then continue down until you have the precision you desire or obtain a root. This is actually how I set a variable resistor in a lab experiment and that's how I know this algorithm works (as assuming anything beyond monoticity in the experiment's relationships wouldve resulting in a sort of circular reasoning).
This is I believe similar to an array searching algorithm. Another method would be binary search (check the middle element and keep progressively cutting tge search region in half).
This is in fact just me giving a couple simple methods. Rootfinding is a large area if study and I would suggest reading about numerical methods in general to see if a quick preferred method jumps out at you. Tricks are nice but they to be greedy algorithms meaning that they can be incorrect at the gain of fast computation.
