A power of a power

There's a second way to group a long product. Take 4⁶, six 4s. Group them as two threes: (4 × 4 × 4) × (4 × 4 × 4) = 4³ × 4³ = 64 × 64 = 4096. But 4³ × 4³ is 4³ squared, so we can write it (4³)².

Group the same six 4s as three twos instead: 4² × 4² × 4² = 16 × 16 × 16 = 4096, which is 4² cubed, or (4²)³.

Same number, two ways of describing it. Likewise 7⁴ = (7 × 7) × (7 × 7) = (7²)², and 2¹⁰ = (2²)⁵ = (2⁵)².

So the rule is (nᵃ)ᵇ = (nᵇ)ᵃ = nᵃ × ᵇ = nᵃᵇ. A power of a power multiplies the exponents, whereas multiplying powers only adds them. Keep those two apart.

Now the Magical Pond. A pink lotus sits in the middle; the number of lotuses doubles every day; after 30 days the pond is completely covered. On which day was it half covered?

Not day 15: that's the trap. Because it doubles each day, whatever covers the pond on day 30 was half of that the day before. So the answer is the 29th day: fully covered is 2³⁰, half covered is 2²⁹.

Damayanti has two empty ponds, one that doubles and one that triples. She puts a lotus in the doubling pond. After 4 days there are 1 × 2 × 2 × 2 × 2 = 2⁴ lotuses. She moves them all to the tripling pond, and 4 days later there are 2⁴ × 3⁴.

What if she had used the ponds the other way round? Then 1 × 3⁴ × 2⁴: the same. Regrouping shows what it really is: (3 × 2) × (3 × 2) × (3 × 2) × (3 × 2) = (3 × 2)⁴ = 6⁴. So mᵃ × nᵃ = (mn)ᵃ when the exponents match and the bases differ, and in the same way mᵃ ÷ nᵃ = (m/n)ᵃ.