Writing 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 is tiring. Writing 2¹⁰ is not. Exponents are shorthand — and they come with a small set of laws that do all the work.
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In an, the base a is what you multiply, and the exponent n is how many times.
⚡ Powers of 2 and 3
| n | 2ⁿ | 3ⁿ |
|---|---|---|
| 1 | 2 | 3 |
| 2 | 4 | 9 |
| 3 | 8 | 27 |
| 4 | 16 | 81 |
| 5 | 32 | 243 |
| 6 | 64 | 729 |
| 10 | 1024 | 59049 |
Two traps. First, 2³ is 8, not 6 — you multiply, not add. Second, anything to the power 0 is 1: 5⁰ = 1, 100⁰ = 1, even (−7)⁰ = 1.
Five laws cover almost every question. Learn what each one does to the exponents.
📜 The five laws
| Law | Rule | Example |
|---|---|---|
| Multiply — add the powers | aᵐ × aⁿ = am+n | 2³ × 2⁴ = 2⁷ = 128 |
| Divide — subtract | aᵐ ÷ aⁿ = am−n | 3⁵ ÷ 3² = 3³ = 27 |
| Power of a power — multiply | (aᵐ)ⁿ = amn | (2²)³ = 2⁶ = 64 |
| Zero power | a⁰ = 1 | 9⁰ = 1 |
| Negative power — flip it | a⁻ⁿ = 1 / aⁿ | 2⁻³ = 1/8 |
The laws only work with the same base. 2³ × 2⁴ = 2⁷ ✅, but 2³ × 3⁴ cannot be combined — work each out separately: 8 × 81 = 648.
Very large and very small numbers are written as a × 10n, where a is between 1 and 10.
🔭 Ordinary ↔ standard form
| Ordinary | Standard form |
|---|---|
| 5 90,000 | 5.9 × 10⁵ |
| 3 20,00,000 | 3.2 × 10⁷ |
| 0.000 07 | 7 × 10⁻⁵ |
| 0.000 000 25 | 2.5 × 10⁻⁷ |
Which way does the sign go? A big number gets a positive power; a number smaller than 1 gets a negative power. Count how many places the decimal point moves — that is the power.
12 questions on exponents and standard form. Apply one law at a time. 🍀
