Someone recently asked what a zero exponent would mean in the real world. Here is my basic response.
The important method of research is to ask and answer questions, and the important questions to ask are “how did we get the concept?” and “what does it mean?” We should look for things in the world where we take exponents. Where did we start? As far as I know, people started in history with geometry, with squares and cubes. What did we do from there? Include natural numbers as exponents. What did we do from there? Include zero as an exponent. What did we do from there? Include negatives, fractions and irrational numbers as exponents.
But let’s consider squares and cubes. We would multiply a length by a length to get an area; this is why we “square,” and from where the name comes. We would multiply a length by a length by a length to get a volume; this is why we “cube,” and from where the name comes. So a cube is a volume, a square is an area, a first power is a line, and so we could define a zero power to be a what? A point. (Considering 5^0 meter to not be the same as (5 meter)^0, just as they are different for the square or cube.)
The person suggested thinking about the number of apples to the zero power. Apples are not a good example, as I’m thinking. Why would we square or cube or otherwise exponentiate the number of apples?
Better examples would be things we can square or cube, or situations where we do so. One good example is compound interest (and continuous interest). We find that we can multiply a dollar amount by a factor, e.g., (1 + 0.08/12), which would be 8% interest compounded monthly, to get a dollar amount for next year. We can multiply that by the same factor to get a dollar amount for two years out. When we look, we find we could’ve multiplied the original amount by the factor squared, (1 + 0.08/12)^2, to get a dollar amount two years out. And so on for three or four years: (1 + 0.08/12)^3 and (1 + 0.08/12)^4, respectively.