I don't think that's the case. As baddox says, it's trivial to show it is true, especially using simplified definitions for exponentiation (i.e. sticking with integer or perhaps rational exponents), but demonstrating truth doesn't tell you about the "reason".
Is the question about a philosophical position as to how mathematics relates to God? A "reason" seems to imply a purpose.
Being able to explain proofs intuitively is a valuable way to check how deeply you know them.
In this case, the reason that when different powers of the same quantity are multiplied together their exponents are added is because powers are short hand for a series of multiplications:
2^4 == 2 * 2 * 2 * 2
When you multiply 2^4 * 2^4, that is short hand for:
2^4 * 2^4 == 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 == 2^8
Of course you can also prove this using algebra, but the intuitive explanation is IMO more useful for building understanding.
That works for positive integer bases and exponents, but try giving an "intuitive proof" with irrational exponents. Most things in math, even seemingly obvious things in arithmetic, require a lot of shared background knowledge (at least propositional logic, basic set theory, and a construction of the natural numbers) for two people to even converse formally.
It still works with irrational exponents (start with fractional numbers and work towards that). It also works with imaginary exponents. It works because the power notation is short hand. But that wasn't the point of my reply. The point was that stating things multiple ways assists us in understanding. Does this not match your experience?
It still works with irrational exponents (start with fractional numbers and work towards that).
So why e^pi * e^e = e^(pi + e)? Yes, it follows from the fact that it works for rational numbers, but in order infer this, you'd need to prove the continuity of exponential function, which is nontrivial at best.
Of course, if you define a^x to be the unique continuous function f: R -> R, such that f(1) = a and f(a)f(b) = f(a+b), as soon as you proved the existence and uniqueness of this function, this follows straight from definition.
There are also different definitions of exponential functions, like exp(x) = lim n->inf (1+x/n)^n, or exp(x) = sum_{n=0}^inf x^n/n! . How easy it is to prove now that exp(pi)exp(e) = exp(pi+e) ?
I would say that "reason" in mathematics is akin to "motivation" for a definition.
In this particular case, the property a^x a^y = a^(x+y) (plus some very weak technical condition, like Lebesgue measurability) uniquely defines exponential functions.
So, in hindsight, you can think of exponentials as arising in the classification of homomorphisms from the additive group to the multiplicative group of reals.
It actually goes deeper than that. You can extend the reasoning to complex numbers (as everyone knows), to matrices, to Lie algebras, and probably beyond.
Is the question about a philosophical position as to how mathematics relates to God? A "reason" seems to imply a purpose.