As a math professor, I think this is a great question. Students learn that math is about manipulating formulas and equations, or about excessive formalities. But being able to explain simple arithmetic facts in clear and plain English is often neglected, and is of the utmost value.
My main problem is the vagueness in wording (which might be attributable to the lack of formalization in mathematics in 1869). What does "reason" mean? Is it asking for a proof? And if so, what axioms and lemmas are you allowed to use? Are we talking about integer bases and exponents (things get much more complicated with rational and irrational exponents)? If you're allowed to assume the definition of exponentiation, then the behavior of multiplied exponents probably follows almost trivially.
To me, this question is equivalent to asking the "reason" that 2 plus 2 equals 4. Everyone "knows why," and understands it pretty well (and could even give an intuitive "proof" by counting), but the question is poorly specified.
Back in those days they would have used slide rules and understood logarithms very well (which they used for multiplication by adding logarithms, essentially). So they may have just answered that to multiply values is to add their logarithms and exponentiate. If the logarithm is taken to the common base, the logarithms are given by the respective exponents.
The generalization of exponentiation makes the "multiply N times" explanation fail.
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.
Is the question about a philosophical position as to how mathematics relates to God? A "reason" seems to imply a purpose.
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.
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) ?
2^2 == 2 * 2
2^3 * 2^2 == 2 * 2 * 2 * 2 * 2 == 2^5
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.
In fact, I was impressed at how open-ended these questions were. For instance: "Leonidas, Pausanias, Lysander" is about as open-ended as you can get.
I for one, would just create a list of interesting anagrams. (eg: Paranoia Saleslady Snide Sun) I figure it would show my moxie.
http://dictionary.reference.com/browse/reason
1. a basis or cause, as for some belief, action, fact, event, etc. ...
3. the mental powers concerned with forming conclusions, judgments, or inferences.
So, its the difference between "reasoning", which we do in math and logic all the time, and "belief" or "motive". That is "reason" did indeed have a specific mathematical meaning at that time, and it still does today.
One might also google "mathematical reasoning"...
I feel like anybody with a basic understanding of what an exponent represents should be able to explain why you add exponents.
It's the same as libraries in programming, really. You don't need to understand how a library works, you need to know what it does and how to use it. And in many cases, that's perfectly fine. After all, who would have the time to study the source code[1] of all libraries they use? You'd study that if you are interested, noticed a bug or need to know some specific detail of the implementation, not just because you need to use it.
[1] Not to mention those unfree binary libraries which you couldn't inspect even if you wanted.
I don't think your example regarding libraries really applies well here. I think a more appropriate example would be knowing how to do a few things with a library without a real ( or any ) understanding of the language its based upon.
I'm no mathematician, nor am I a (professional) programmer for that matter, yet I've seen real benefits going back and really learning some of those topics that were kinda passed over in high school ( so what exactly is sine doing to my numbers? Taylor series? Fourier series? Optimization functions?). Besides, math teaches you to think, memorization doesn't.
=
(a*a*a*a*a) * (a*a*a*a*a*a*a)
^ ^
n times m times
X^N * X^M
= N copies of X, multiplied by M copies of X
= N+M copies of X multiplied together
= X^(N+M)x^n = exp((log x)n). By definition, exp(n+m)=exp(n)exp(m). By definition, a(b+c) = ab + ac (for the log x thing). QED.
By the way, (the infamous calculus book) Baby Rudin has the poor reader show this property holds in exponentiation for reals, starting with integers and via rationals, as an exercise on its first chapter. Insane difficulty for me, even though the author practically holds your hand along the way! Cool read, though.
a^(n+1)=a^n * a
a^(n+2)=a^n * a^2
a^(n+3)=a^n * a^3
a^(n+0)=a^n * a^0
therefore
a^(n+m)=a^n * a^m
I personally like cousin_id's the best, as it is (IMHO) the simplest.