# Calculate the exact value f l o o r ( <mrow class="MJX-TeXAtom-ORD"> &#x3C

Calculate the exact value $floor\left({\pi }^{k}\right)$ without actually calculating the huge value of ${\pi }^{k}$.
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lofoptiformfp
The only way I know of to speed up exponentiation modulo an integer $n$ (compared to general exponentiation) is using the fact that everything is integers to reduce the intermediate results modulo $n$. As the intermediate results in your case aren't integers, you can't do that.
The only transformation I can see of your expression is:
$⌊{x}^{k}⌋\phantom{\rule{1em}{0ex}}\left(\mathrm{mod}\phantom{\rule{0.333em}{0ex}}n\right)=⌊{x}^{k}\phantom{\rule{1em}{0ex}}\left(\mathrm{mod}\phantom{\rule{0.333em}{0ex}}n\right)⌋$
and that doesn't make the computation any easier.