A review of the rule that $n^0$ is always 1, when n is not 0, has made me question if all math rules can be visualized or conceptualized with a general intuition gained from viewing the natural world. It seems that me that there is no natural representation of this rule or a representation that can be conceptualized in terms gained from experiencing reality. I have come across three methods of justification that are used to determine that $n^0=1$. One uses a somewhat altered, or added to, definition of exponents, the other relies on maintaining internal consistency with other math rules, and the last relies on a pattern of division seen with single terms exponentiated. I’ll list the justifications now:
The second justification has to do with assuming that the definition of exponents always starts with a multiplication of 1. Therefore $x^3=1*x*x*x$ and $x^0=1$
The last justification tries to justify the rule from a pattern: $3^3=27$, $3^2=9$, $3^1=3$. Each time the following result is the former result divided by 3. If we keep going, we get $3^0=1$.
None of these methods make very much conceptual sense because none of the justifications are attempts to show conceptual justifications, instead they are abstracted away from real life examples and intuition. If we think about exponentiation as repeated multiplication, similar to how multiplication is repeated addition, I cannot think of an everyday situation which would repeat multiplication 0 times and get 1. As a result, I cannot grasp an intuitive sense of the rule. With multiplication by 0, it is easy to see that when you repeat addition on X 0 times, you get 0.
So, my question is, is there a more intuitive explanation that I am missing or not understanding? If not, then I feel like I have been treating my math studies in the wrong way. I have been trying to intuitively understand mathematical concepts in the same way a philosopher mathematician might have conceptualized them early on in their brains. Is this the wrong way to go about thinking about math? Should I instead just treat math as a series of arbitrary rules that are built upon each other to create a system? Edit: Or as LittleO said, "convenient rules".
Edit 1: Called the three justifications "rules" on accident.
Also thank you to the kind admin/moderator who helped clean up the math formatting.