Not really; a fractional exponent n yields the quantity one would have to multiply 1/n times to return the original value. Multiplication an integer number of times could be seen as a special case of a broader concept of "fractional" multiplication (much like the gamma function (Γ(n)) extends the discrete factorial to a continuous domain).
How do you explain irrational exponents this way, for example? What about complex exponents?
Indeed you can extend the special case to the continuous domain -- but then the definition is expanded as well.
I still think "multiply N times" is just a special-case, and as such, not usable as a definition -- let alone an explanation of why we can add exponents in the general case.
The generalization of exponentiation makes the "multiply N times" explanation fail.