how can you calculate 2 to the 4.31 power without a calculator.?

Answer 1

You would need a table of logs. You multiply the log of 2 by 4.31 and then take the antilog.

Answer 2

You probably can't do it exactly.
But you can get a pretty good approximation.

You can calculate 2^4 (it's 16), and 2^5 (it's 32), so the answer has to be between 16 and 32.

And you know that the answer is closer to 16 than to 32.
You might think that it would be 31% of the way from 16 to 32 (because of the .31 in 4.31). But since exponential functions grow more rapidly as the exponent increases, it will be LESS than 31% of the way from 16 to 32 (i.e., it will be less than
16 + 31% of (32 - 16), which would be about 21.

So the answer is a little less than 21. You might use 20 as an approximation.
And 20 is very close to the right answer.