multiplying same-base values means you add the indices, right?
this means x^2 * x^-1 = x^1
if we let y = x^-1 then x^2 * y = x
x * y = 1
y = 1/x
therefore x^-1 = 1/x

I feel like a lot of math is defined like that because it works well in the broader system, and the properties of the system are useful.

You can have a different math system that doesn't allow reals in exponents, but then certain problems might require inventing a whole lot more complicated systems to work around it.

Look up Lunar Arithmetic on Wikipedia for a different (simpler) mathematical system for multiplication and addition. It has some cool properties, but those properties aren't generally the ones we want for daily usage math.

I think is is difficult to explain why we use certain conventions without also explaining the paths not take. Historically some people might have taken those paths and bumped into issue. It is very cool to explore alternatives, but I think most highschool teachers aren't going to be familiar with them.