So, say you have a really huge whole number like $5^{2000}$ and an irrational number like $\sqrt(5)$.
If you were two multiply the two would you get an even or odd number after rounding to the nearest whole number? $$5^{2000} * \sqrt(5)$$
I realize I could input this into a calculator and have it tell me, but is there a general why to know whether the answer would be even or odd?
I'm thinking it has something to the number of digits in the whole number, but I can't think of a way to guaranteed method(I've tried looking this up ): ). How many digits of the irrational number would I need?
I presume you are asking about the parity of the whole number you get by rounding or truncating $5^{2000}\sqrt 5$. The issues are the same whether you round or truncate, though the number you get may be different and the answer may be different.
$5^{2000}$ is a number of $\lceil 2000 \log_{10}5\rceil=1398$ digits. In fact, it is rather close to $10^{1398}$ If we want to know $5^{2000}\sqrt 5$ to within a possible error of $\pm 0.5$ we need to know $\sqrt 5$ to within $\pm 0.5\cdot 10^{-1398}$, which means you need $1399$ digits of $\sqrt 5$ past the decimal point.
There are perverse cases where the error you can tolerate is much smaller than $\pm 0.5$. It happens when the exact product is very close to the breakpoint between two rounded values. If we are truncating and the correct answer might be $2.999999$ and might be $3.000001$ a difference of only $0.000002$ is enough to change the result. The nature of truncating or rounding makes it impossible to avoid this.