Sure, I can explain this pretty easily! :)
All right, so you should already know most of this, but I'm going to build this up in order to show the sequence of steps.
First off, the definition of exponentiation:
an = a × a × ... × a
That is, exponentiation is repeated multiplication. Next, a useful property:
ab × ac = ab+c
Proof:
(A) ab = a × a × ... × a (b times) (definition of exponentiation)
(B) ac = a × a × ... × a (c times) (definition of exponentiation)
(C) ab × ac = (a × a × ... × a (b times)) × (a × a × ... × a (c times)) (substitution)
(D) ab × ac = (a × a × ... × a (b+c times)) (associativity of multiplication)
(E) ab × ac = a(b+c) (definition of exponentiation)
But what if there's a negative number in the mix? We can't count
a -2 times, after all, so we can't write it out as repeated multiplication. But let's apply (D) from the above proof...
(F) ab-c = (a × a × ... × a (b-c times)) (substitution)
(G) ab × a-c = (a × a × ... × a (b-c times)) (as proved above)
(H) (a × a × ... × a (b times)) × a-c = (a × a × ... × a (b-c times)) (definition of exponentiation)
Okay, so let's just think about this... What could we possibly substitute into
a-c to make this equation true? We need to take away some
a's. Let's solve for it:
(I) (a × a × ... × a (b-c times)) = (a × a × ... × a (b times)) × a-c (rearranging for formatting)
(J) (a × a × ... × a (b-c times)) = a-c (divide both sides)
____(a × a × ... × a (b times))
(K) 1/(a × a × ... × a (c times)) = a-c (cancellation)
(L) 1/(ac) = a-c (definition of exponentiation)
And there you go -- the property you're looking for!
Said in plain English: Taking a number to a negative exponent is a way to take away multiplications by the number instead of a way to do more multiplications by it. And we take away multiplications by using division!