Which explanation best justifies why multiplying powers with the same base will mean that you can add the exponents, while dividing them will mean that you can subtract the exponents?

Exponents count repeated factors. When multiplying, the factor lists combine, so the exponents can be added together like so: a^m × aⁿ = a^m+n. When dividing, the matching factors will cancel, so the exponents will subtract like so: a^m / aⁿ = a^m-n.

Multiplication and division cannot be explained all together by using factors. Adding or subtracting exponents is just a coincidence with no real justification from repeated multiplication.

Multiplication adds exponents because the exponents themselves are multiplied together. Division subtracts exponents because dividing them inverts the denominator before multiplying.

Exponents represent logarithms. Multiplication adds the logs, and division subtracts the logs. This is the only correct justification.