Core answer: aⁿ means a multiplied by itself n times: 2¹⁰ = 1,024. Five laws govern everything: aᵐ·aⁿ = aᵐ⁺ⁿ, aᵐ/aⁿ = aᵐ⁻ⁿ, (aᵐ)ⁿ = aᵐⁿ, a⁰ = 1, a⁻ⁿ = 1/aⁿ. Fractional exponents are roots: a^(1/2) = √a, a^(1/3) = ∛a, 8^(2/3) = (∛8)² = 4.
The five laws with instant examples
| Law | Formula | Example |
|---|---|---|
| Product | aᵐ·aⁿ = aᵐ⁺ⁿ | 2³·2⁴ = 2⁷ = 128 |
| Quotient | aᵐ/aⁿ = aᵐ⁻ⁿ | 5⁶/5² = 5⁴ = 625 |
| Power of power | (aᵐ)ⁿ = aᵐⁿ | (3²)³ = 3⁶ = 729 |
| Zero | a⁰ = 1 (a≠0) | 999⁰ = 1 |
| Negative | a⁻ⁿ = 1/aⁿ | 2⁻³ = 1/8 = 0.125 |
Why a⁰ = 1: the quotient law demands aᵐ/aᵐ = a⁰, and anything over itself is 1.
Fractional exponents = roots
x^(m/n) = (ⁿ√x)ᵐ. So 27^(2/3) = (∛27)² = 3² = 9. Take the root first (small numbers) then the power — sanity preserved.
Exponential growth: why it surprises everyone
- Doubling: bacteria doubling every 20 min: after 12 h (36 doublings) one cell becomes 2³⁶ ≈ 68.7 billion.
- The chessboard: 2⁶³ grains on the last square ≈ 9.2×10¹⁸.
- Paper folding myth: 0.1 mm paper folded 42 times reaches the Moon (0.1 × 2⁴² mm ≈ 440,000 km). Physically impossible past ~7–8 folds — the math outruns the material.
Computing example. 2¹⁰ = 1,024 ≈ 10³, which is why KB/MB/GB confuse buyers: 1 KB = 1,024 B (binary) vs 1 kB = 1,000 B (SI) — a 2.4% gap that grows to ~7% at GB scale.
Common mistakes and myths
- aᵐ·aⁿ = aᵐⁿ — no, exponents add: 2³·2⁴ = 2⁷, not 2¹².
- (a+b)ⁿ ≠ aⁿ + bⁿ — (2+3)² = 25, not 4 + 9 = 13 (the "freshman's dream").
- 0⁰ — undefined/indeterminate in most contexts; don't assume 1 blindly in limits.
- (−2)² vs −2² — the first is 4 (base is −2); the second is −4 (exponent binds before the minus). Parentheses decide.
- Thinking exponentials are fast but bounded — exponential beats every polynomial eventually: 2ⁿ passes n¹⁰⁰ near n ≈ 1,000. That "eventually" is the engine of both compound interest and algorithm blowups.