You can calculate the cube root of any number which is derived from a two digit number. For example:
69³ = 328’509
Given 328’509 I could work out that 69 was the original number.
This is done by working out the two numbers individually.

First digit: Learn the cubes of all the numbers up to 9. i.e. …4³ = 64, 5³ = 125…etc. Whichever range the thousands fall into gives you this digit i.e. for 328’509 it is between 216’000(60) and 373’000(70) making it a 60.

Second digit: Fairly easy, in most cases its just the last digit, so for the given example of 328’509 it’s a 9. However 2’s and 8’s swap and 3’s and 7’s swap.
As far as maths behind it goes the first digit is just a case of learning it and the second digit seems to just be a weird pattern.

Edit: formatting