So, I found that you can work out cube roots between $1$ and $100$ using this method:
For example the number: $185193$
If the last digit $x$ is $2$, $8$, $3$, or $7$, the last digit in the answer should be $10 - x$; if otherwise, then $x$. In this case:
$$10-3 = 7$$
We now got $\_7$. Remove the last three digits so that we get only $185$. Then, check in this list:
$$ 1^3 = 1 \\ 2^3 = 8 \\ 3^3 = 27 \\ 4^3 = 64 \\ 5^3 = 125 \\ 6^3 = 216 \\ 7^3 = 343 \\ 8^3 = 512 \\ 9^3 = 729 $$
Check which two cubes that number is between, the lowest one should be the first digit in the answer:
$$5^3 < 185 < 6^3$$
Therefore, we get:
$$57$$
I found this method in this video: https://www.youtube.com/watch?v=Al7PllobwR4
But how does this method really work, and why?