First, the one idea everything rests on: an index just tells you how many times to multiply the base by itself. So 3⁴ means 3 × 3 × 3 × 3. The little raised number (the index, or power) is a counter — it's counting the factors. Hold onto that and the rest is easy.
Multiplying: Just Add Up the Factors
What's 3² × 3³? Write it out the long way: (3 × 3) × (3 × 3 × 3). Now count the 3s — there are two, then three, so five of them in a row. That's 3⁵. You didn't really do anything clever; you just pushed the two piles of 3s together and counted. That's why aᵐ × aⁿ = aᵐ⁺ⁿ — the indices add because the factors line up end to end.
Dividing: the Factors Cancel
Now 3⁵ ÷ 3². Written out it's five 3s on top, two 3s on the bottom. Each 3 on the bottom cancels a 3 on the top — that's two pairs gone — and what survives is three 3s, i.e. 3³. Same base, you subtract: aᵐ ÷ aⁿ = aᵐ⁻ⁿ. Dividing is the undo of multiplying, so it makes sense the indices go the other way.
A Power of a Power: Groups of Groups
What about (3²)³? That's three copies of 3² multiplied together: (3 × 3) × (3 × 3) × (3 × 3) — three groups of two 3s, so six 3s, which is 3⁶. Three lots of two is 2 × 3 = 6. So this time you multiply the indices: (aᵐ)ⁿ = aᵐⁿ.
And the Sneaky One: Anything to the Power Zero
Here's where a⁰ comes from, and it's lovely. Take 3³ ÷ 3³. By the dividing rule that's 3³⁻³ = 3⁰. But it's also just a number divided by itself — and anything divided by itself is 1. So 3⁰ = 1. It's not a special exception someone made up; it's forced on us by the very same law. Try it in the toy with the a⁰ tab — all the factors cancel and 1 is what's left.