there is really fun way to mentally calculate cube root of numbers of form n³ where n is in range of [10, 99].
first you have to memorize numbers from 1 to 9 cubed.

13	1
23	8
33	27
43	64
53	125
63	216
73	343
83	512
93	729

you have to remember that numbers
10, 20 ... 90 cubed are just numbers 1, 2 ... 9 cubed times 1000.
so 10³ = 1000, 20³ = 8000 and so on

now we can observe something interesting
if we look at last digits of second column of table above we can see that they don't repeat (meaning that there exist bijection from set of numbers 1 to 9 to last digits of their cubes)

another helpful fact is 1, 4, 6 and 9 correspond to themself and that 2 and 8 are switched and 3 and 7 are switched.

observe that thir sum is 10 so you can only remember that 2, 3, 7 and 8 don't correspond to themself. to recover what they correspond to you just have to subtract them from 10. for example to see what 3 corresponds to you just calculate 10 - 3

having all that we can attack our cube root problem. there are 2 stages to this trick

first you have to find the biggest number from 10³ to 90³ that is smaller than the number of which we want to calculate cube root. that is our tens number

secondly we look at last digit of our number and see to what it corresponds to in above table (we look at ones digits of numbers in 2 column and see what they correspond to)

let's see an example

we cubed some number and got 830584. we see that biggest number from range [10^3, 20^3, ... 90^3] smaller than 830584 is 90^3 so we know that number cubed was 90-something. then we look at last digit of 830584 and we see that it is 4 so we know that number cubed was 94!

i learned this trick from Howie Hua on tiktok so go and follow him!!!

answering question that brought you here - above widget is place for you to train newly learned skill. you just have to put value of expression on the left in the text field and press enter