How to polish a turd?

I assume when you mean situation, you mean interview question.

Think what the purpose of the question is. Does the interviewer know how hard this is? Assume yes.

So he asks you a near impossible numerical question. Why would he do that? To find out what you do when faced with a situation you cant see a way out of. Do you instantly give up, do you panic and try random stuff, or do you dig deep into what you know is relevant material and try something meaningful.

Trazer's best interview advice for brainteaser style questions in an interview is:

Nothing is as hard as it looks, nothing is as easy as it seems.

Given your example Cube roots are surprisingly easy to do. You can deduce the order of magnitude, and the last digit (or at least reduce it to a choice of a few). Without any further common sense filtering you have at most 20 numbers to try (if you cant tell whether cube root of 1000 is nearer to 10 than 99 then you have bigger issues.

With a few trial and errors you can easily figure out what it is.

example you're aiming for the cube root of 857375, 90^3 = 729000.

That's pretty easy to do in your head. You know it cant be into the 100's since 100^3 = 1,000,000 and it ends in a 5. so 95^3 is a pretty safe bet.

Problems come up if the cube root isnt an integer, but that case is impossibly rare.

The last 4 digits of x^81 is interesting. whenever i see 81 in any logic puzzle at all i instantly think Thats a mighty coincidental number - 3^4.

so x^81 = x^ 3 ^ 4. so cube your x, and then do the multiplication out on paper, discarding every number that goes into the 10^4 column.

If it's just the last digit, you can often write out the pattern , and work it out from the index (powers of 2 -5 i would do this for).

Key things to avoid:

DONT answer instantly. Count to 10 at least, then answer in your head, then sense check it Panicking/saying I dont know.

Things to do:

React, clarify the question out loud, to ensure you have understood it, Say it's a pig of a question, you weren't expecting something like that etc.