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.

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.

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.