Powers & roots

Fold a sheet of paper, then again, and again: each fold doubles its thickness. After just 42 folds the stack would in theory be thicker than the distance to the moon. This kind of growth through repeated multiplication is behind storage sizes, viruses, interest, and chain letters. Powers are the shorthand for it.

Powers: repeated multiplication

Just as 3 × 4 is shorthand for 4 + 4 + 4, 2³ is shorthand for 2 × 2 × 2 = 8. The number in normal size is called the base, the small raised number is the exponent: it counts how often the base appears as a factor. Watch out for a classic trap: 2³ is not 2 × 3 = 6, it is 8. The difference gets more dramatic with every step: 2¹⁰ is already 1024.

Square numbers and roots

Powers with exponent 2 are called square numbers, because they describe squares: a grid of 5 rows with 5 squares each has 5² = 25 squares. The square root goes the other way: √25 asks which positive number multiplied by itself gives 25, namely 5. Taking a root undoes squaring, just as dividing undoes multiplying. Knowing the square numbers up to 12² = 144 makes you really fast here.

Powers of ten: big numbers under control

Powers of ten are especially convenient: the exponent simply counts the zeros. 10³ = 1000, and 10⁶ is a million. That lets you write huge numbers compactly: the Earth is about 150 million kilometres from the sun, so roughly 1.5 × 10⁸ kilometres. Your storage space thinks in powers too: kilo, mega, and giga stand for 10³, 10⁶, and 10⁹ bytes.