The other side of powers

Powers can shrink as well as grow. Picture a line 2⁴ = 16 units long and rub out half of it. What's left is (2 × 2 × 2 × 2) ÷ 2 = 2 × 2 × 2 = 2³ = 8 units.

Halve it again: 2⁴ ÷ 2² = 2² = 4. A third time: 2⁴ ÷ 2³ = 2¹ = 2. Each halving cancels one 2 from the top.

So 2⁴ ÷ 2³ = 2⁴⁻³ = 2¹, and in general nᵃ ÷ nᵇ = nᵃ⁻ᵇ: dividing powers subtracts the exponents. (n must not be 0, because we can't divide by 0.)

Now push the rule somewhere new. What is 2⁰? If the rule is to keep working, 2⁰ = 2⁴⁻⁴ = 2⁴ ÷ 2⁴ = 16 ÷ 16 = 1.

So x⁰ = 1 for any x that isn't 0. That isn't a random choice: it's the only value that keeps nᵃ ÷ nᵇ = nᵃ⁻ᵇ true.

Push further. Halve the 16-unit line five times: 2⁴ ÷ 2⁵ leaves half a unit. By the rule that's 2⁴⁻⁵ = 2⁻¹. So 2⁻¹ = 1/2. Halve it ten times: 2⁴ ÷ 2¹⁰ = 2⁻⁶, and written out that's 1/2⁶ = 1/64. So a negative exponent means 'one over' (a reciprocal): 10⁻³ = 1/10³, 7⁻² = 1/7². Flip the number, but keep the base's own sign: (−4)⁻³ = 1/(−4)³.

It works backwards too: 1/10⁻³ = 1 ÷ (1/10³) = 10³. So n⁻ᵃ = 1/nᵃ and nᵃ = 1/n⁻ᵃ, for n ≠ 0.

The book lays the powers of 4 along a power line: 4⁸, 4⁷, …, 4¹, 4⁰, 4⁻¹, 4⁻², that is 65536, 16384, …, 4, 1, 1/4, 1/16. Each step left multiplies by 4 and each step right divides by 4. On it you can read straight off that 4⁷ is 4² = 16 times larger than 4⁵, because 4⁷ ÷ 4⁵ = 4².