I realized something the other day that's got my mind in a knot again. Given some constant $ n $, I know that: $$ n^2 = n \cdot n$$ $$ n^3 = n \cdot n \cdot n$$ And so on so forth. I also know that: where, $ n^{1/2} = k $, then: $ k \cdot k = n $ and so on and so forth. Things get a little complicated when $ n^{2/3} = k $, but with some work I understand that $ n \cdot n = k \cdot k \cdot k $. An intuitive explanation might be "the length of one side of that square is equal to the length of one side of that cube." Higher dimensions, of course, break this "intuitiveness" but I can still process mathematically what these rational powers mean.

But what does it mean when you're given $ n^\pi $? How would I visualize this? Is it really as simple as just approximating to the degree of accuracy you need?

$$ n^{31415} = k^{10000} $$

What's a more rigorous definition (or alternative explanation) of $ n^\pi $ than my understanding of powers mentioned above?

I ask this question because I read this thread: Real Numbers to Irrational Powers