Multiplying powers

The book sets a riddle in verse: 'Three daughters with curious eyes, Each got three baskets, a kingly prize. Each basket had three silver keys, Each opens three big rooms with ease. Each room had tables, one, two, three, With three bright necklaces on each, you see. Each necklace had three diamonds so fine… Can you count these stones that shine?'

Draw it as a branching tree. The king branches into 3 daughters; each into 3 baskets (3 × 3 = 9); each into 3 keys (27); each into 3 rooms. That's four branchings of 3, so the number of rooms is 3⁴.

Work it out by multiplying step by step: 3 × 3 = 9, 9 × 3 = 27, 27 × 3 = 81. So 81 rooms.

Keep going for the diamonds: rooms, then tables, then necklaces, then diamonds. That's seven branchings in all, so there are 3 × 3 × 3 × 3 × 3 × 3 × 3 = 3⁷ diamonds.

Do we have to start again from 1? No. Split the seven 3s: 3⁷ = (3 × 3 × 3 × 3) × (3 × 3 × 3) = 3⁴ × 3³ = 81 × 27 = 2187. We already had 81, so one multiplication finishes the job.

The split doesn't have to be 4 and 3. 3⁷ is also 3² × 3⁵: all that matters is that the counts add to 7. It works with letters too: p⁴ × p⁶ = (p × p × p × p) × (p × p × p × p × p × p) = p¹⁰. Four p's next to six p's is ten p's; that's all there is to it.

So the rule, for counting numbers a and b: nᵃ × nᵇ = nᵃ⁺ᵇ. When you multiply powers of the same base, you add the exponents. (It's only for multiplying: 7⁴ + 7 is not 7⁵.)