I have recently learned how to calculate $a^b$ for all $a,b \in \mathbb{Q}$.
I have noticed that the $a$ in fraction form will always be $a\cdot 10^n/10^n$ where $n$ is $a$’s decimal length. But if $a$ is a big number, it would be painful to write the exponent and to calculate the n-th root (in this case $10^{200}$'th root of $3.1415926\dots$).
Is there any easy/efficient way to calculate this? How might we best approximate $x^\pi$?
Btw I am a 5th grader so please try to explain it well (if possible)!
Thank you.
Bankers have an underestimate of compound interest gained, that will be helpful here. It says, add the percentages. If our percentage interest, goes above $2\times 10^{-198} \%=2\cdot 10^{-200}$ this method shows that the percentage interest gained after $10^{200}$ periods plus deposit, will exceed pi when written as a decimal. If estimated to 200 places past the decimal ( 201 significant figures) if we knock off that 201st significant figure we get a near perfect estimate to 200 digits of 1 . As to the title question (which isn't the same), You could, but you'd be wasting your time, if you know how to program it.
Note:
This estimate is very precise, but not very accurate.