Exponential notation

Writing 0.001 × 2 × 2 × 2 × 2 × 2 × 2 × 2 in full is tiring. There's a shorthand, and you already know how it starts.

n × n = n², read 'n squared' or 'n raised to the power 2'. n × n × n = n³, 'n cubed' or 'n raised to the power 3'. And it carries on: n × n × n × n = n⁴, 'n raised to the power 4'.

In general, nᵃ means n multiplied by itself a times. So the thickness after 7 folds is 0.001 cm × 2⁷ = 0.128 cm.

Take 5⁴ = 5 × 5 × 5 × 5 = 625. We say 5⁴ is the exponential form of 625.

Numbers like 5ⁿ are called the powers of 5.

And 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 = 2¹⁰ = 1024: the very number that kept turning up when the paper was folded ten more times.

It works for negatives and letters too:

• 4 × 4 × 4 = 4³ = 64;
• (−4) × (−4) × (−4) = (−4)³ = −64;
• (−1) to an even power is 1, to an odd power −1; and 1 to any power is 1;
• a × a × a × b × b is written a³b², read 'a cubed b squared'; a × a × b × b × b × b is a²b⁴.

A warning worth pinning up: 4 + 4 + 4 = 3 × 4 = 12, but 4 × 4 × 4 = 4³ = 64. Repeated adding and repeated multiplying are completely different things.

Exponents also tidy up prime factorisation: 32400 = 2 × 2 × 2 × 2 × 5 × 5 × 3 × 3 × 3 × 3, which in exponential form is 32400 = 2⁴ × 5² × 3⁴: ten factors squeezed into six symbols.