Exponents are shorthand, and every law is just that shorthand written out. If you are ever unsure, expand a small case — 2³ × 2² = (2·2·2)(2·2) = 2⁵ — and the rule proves itself.
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Six laws cover everything. All of them need the same base.
📜 The six laws
| Law | Example |
|---|---|
| aᵐ × aⁿ = am+n | 2³ × 2⁴ = 2⁷ = 128 |
| aᵐ ÷ aⁿ = am−n | 3⁵ ÷ 3² = 3³ = 27 |
| (aᵐ)ⁿ = amn | (2³)² = 2⁶ = 64 |
| (ab)ᵐ = aᵐ × bᵐ | (2×5)³ = 8 × 125 = 1000 |
| a⁰ = 1 (a ≠ 0) | 7⁰ = 1, and 100⁰ = 1 |
| a−n = 1 ÷ aⁿ | 2⁻³ = 1/8 |
A negative index does not make the number negative. 2⁻³ = 1/8, which is positive. What is negative is the power, not the value.
The laws need equal bases. 2³ × 3² is not 6⁵ and not 2⁵ — it is 8 × 9 = 72. Rewrite to a common base where you can: 4³ = (2²)³ = 2⁶.
This is where most Olympiad marks are lost.
➖ Two rules to hold on to
Flip a fraction to kill a negative index. (a/b)−n = (b/a)n. So (3/5)⁻² = (5/3)² = 25/9.
A negative base needs brackets. (−2)⁴ = 16 but −2⁴ = −16. An even power of a negative number is positive, an odd power is negative.
Anything to the power zero is 1 — so 3⁰ + 4⁰ + 5⁰ = 1 + 1 + 1 = 3, not 0 and not 12. This exact question appears in almost every paper.
A number written as k × 10ⁿ where k is between 1 and 10.
🔬 Big and small
| Number | Standard form |
|---|---|
| 5 900 000 000 | 5.9 × 10⁹ |
| 384 000 000 m | 3.84 × 10⁸ m |
| 0.000007 | 7 × 10⁻⁶ |
| 0.00000000016 m | 1.6 × 10⁻¹⁰ m |
Which way does the sign go? A number bigger than 10 gets a positive power; a number smaller than 1 gets a negative power. Count how many places the decimal point moves — that count is the power.
12 Olympiad questions on powers. Match the bases before you touch the indices. 🍀
