Mathematics · Arithmetic
Powers stack multiplication; roots take it back
Try this first
A rumour starts with one person, and every hour each person who knows it tells one new person. After 10 hours, roughly how many people know — closer to 20, or closer to 1,000? Guess before you compute.
Most people guess low, because our intuition adds when the world multiplies. Each hour doubles the count: 1, 2, 4, 8, 16,… and after 10 hours it's 2×2×… ten times over — 1,024. We write that pile-up of multiplications as a power: 2¹⁰. Just as multiplication was a shortcut for repeated addition, an exponent is a shortcut for repeated multiplication.
The notation has two parts: 2⁷ means "the base 2, multiplied by itself exponent-many times" — here, 7 twos: 2×2×2×2×2×2×2 = 128. The reason it feels surprising is in the picture — doubling barely moves at first, then erupts:
The one idea
A power bⁿ is the base b multiplied by itself n times. A root undoes it: the square root √9 asks "what number, squared, gives 9?" — the same inverse move that division was for multiplication.
Roots are the question powers answer
Squaring a number means raising it to the power 2: 3² = 9. The square root runs the film backward — √9 = 3, because 3² = 9. (It's called "square" because 3² is the area of a 3-by-3 square, tying straight back to the grid from lesson 3.) Most roots aren't whole: √2 ≈ 1.414…, an honest number that never settles — but you rarely need it by hand, only to know what it asks.
Three facts about exponents earn their keep early. They're worth recognising more than memorising:
| Expression | Value | Why |
|---|---|---|
b¹ | b | one copy of itself |
b⁰ | 1 | no copies — the empty product |
b⁻¹ | 1 / b | a negative power means "one over" |
2³ × 2² | 2⁵ = 32 | multiplying powers adds the exponents |
That last row is the quiet bridge to the next part of the course: turning multiplication into addition by working with exponents is exactly what a logarithm (lesson 24) does in reverse.
Work one, then finish one
Worked: compute 2⁴. Don't reach for a calculator — just double from 1, four times: 1 → 2 → 4 → 8 → 16. So 2⁴ = 16.
Your turn: a "deep" network stacks layers, and a toy one doubles its neuron count each layer starting from 1. How many neurons in the 8th layer, and what is √64? (Answer: 2⁷ = 128 in the 8th layer; √64 = 8.)
Why this earns a place in your toolkit
Squares and square roots are how machines measure distance. The gap between a prediction and the truth is almost always squared before it's added up — that's the "squared error" a model spends its whole life shrinking, and squaring is what makes every miss count as positive and punishes big misses hardest. The length of a vector, the distance between two data points, the norm that decides whether two things are "close" — all are a sum of squares with a square root on top (the Pythagorean theorem, scaled up). Meanwhile exponential growth, that runaway curve above, is how the cost of an algorithm explodes as data grows, and the reason "it worked on 100 rows" can die on a million. Powers are where size and distance enter the math.
Recall check · no peeking
- What does
5³mean as a repeated multiplication, and what is it? - What question does
√49ask, and what's the answer? - Why is
b⁰ = 1for any base — what's being multiplied? - Where does squaring show up when a model measures how wrong it was?
Explain it back
In one sentence, explain to a friend why doubling ten times lands near a thousand and not near twenty.