Index Laws
Three rules that turn multiplying, dividing and powering powers into adding, subtracting and multiplying small numbers. None of them needs memorising: each one is just counting how many copies of the base you have.
National curriculum KS3: use the index laws for multiplication and division of integer powers; GCSE extends to zero, negative and fractional indices1 · A power is a count
Remember what the index means (see powers and roots): a⁵ is five copies of a multiplied together. Nothing more. Every law on this page comes from writing the copies out, doing something obvious, and counting what's left. Let's agree a picture: each block is one copy of the base a.
2 · Law 1: multiplying powers
a³ × a² is three copies, then two more copies, all multiplied together. That's five copies: a⁵. The indices add. The bases must be the same letter: you're counting copies of one thing.
Not a × rule. a³ × a² is not a⁶. Three blocks and two blocks are five blocks. If you're tempted to multiply the indices, picture the blocks lined up and count them.
3 · Law 2: dividing powers
a⁵ ÷ a² is five copies on top of a fraction and two underneath. Each copy below cancels one above (a ÷ a = 1). Two pairs cancel, three copies survive: a³. The indices subtract.
Slide n past m and watch what happens: more copies underneath than on top. That's where zero and negative indices come from, in section 5.
4 · Law 3: a power of a power
(a³)² means "a³, squared": two copies of a³. Each a³ is itself three copies of a, so you have 2 groups of 3, which is 6 copies: a⁶. The indices multiply. This is the one law where multiplying is correct, because you have groups of groups.
Side rules that follow
| Rule | Statement | Why |
|---|---|---|
| Power of a product | (ab)ⁿ = aⁿbⁿ | (ab)³ = ab × ab × ab = a·a·a × b·b·b. The power lands on each factor, including a number: (2x)³ = 8x³. |
| Power of a fraction | (a/b)ⁿ = aⁿ/bⁿ | Same idea on top and bottom: (x/3)² = x²/9. |
| Coefficients | 3a² × 4a⁵ = 12a⁷ | Multiply the numbers normally, apply the law to the letters: 3 × 4 = 12, a² × a⁵ = a⁷. |
| Different letters | a²b × ab³ = a³b⁴ | Each letter gets its own count: a² × a = a³, b × b³ = b⁴. |
5 · Zero and negative indices
Law 2 with n bigger than m forces something new. a³ ÷ a³ = a⁰ by the law, but it's also three copies cancelling three copies, which is 1. So a⁰ = 1. And a² ÷ a⁵ = a⁻³ by the law, but drawn out it's two cancelling two with three left underneath: 1/a³. So a⁻ⁿ = 1/aⁿ: a negative index means "on the bottom".
Negative index ≠ negative number. 2⁻³ = 1/8, which is positive. The minus moves the power to the denominator; it doesn't make anything negative. And a⁰ = 1, not 0: zero copies multiplied is the starting point for multiplication, which is 1.
6 · Fractional indices (GCSE)
What could a½ mean? Use Law 1: a½ × a½ = a½+½ = a¹ = a. So a½ is the number that multiplies by itself to give a. That's the square root. Likewise a⅓ is the cube root, and a⅔ = (a⅓)² is "cube root, then square".
Order matters for your sanity, not for the answer: root first (small numbers), then power. 8⅔ as (∛8)² is 2² = 4; as ∛(8²) it's ∛64 = 4. Same answer, but one route goes through 64.
7 · Spot the error
Each line below is either right or wrong. Decide, then click. These are the mistakes that appear on exam scripts every year.
8 · Where index laws live
Standard form
(3 × 10⁸) × (2 × 10⁵) = 6 × 10¹³. Multiplying huge numbers becomes 3 × 2 and 8 + 5. Dividing is 8 − 5.
Computing: binary sizes
2¹⁰ bytes in a KiB, 2²⁰ in a MiB. A MiB divided by a KiB is 2²⁰ ÷ 2¹⁰ = 2¹⁰ = 1024. A 2⁵-bit address space squared is 2¹⁰ possibilities… law 3.
Algebra ahead
Differentiation (the first page in this series) rests on xⁿ → nxⁿ⁻¹. Simplifying algebraic fractions like 6x⁵/2x² = 3x³ is Law 2.
Compound growth
1.05¹⁰ × 1.05⁵ = 1.05¹⁵: fifteen years of 5% growth is ten years then five. (1.05¹²)² is 24 years.
Units
Area m², volume m³. Speed m/s = m·s⁻¹; acceleration m·s⁻². Negative indices are how scientists write "per".
Logarithms (A-level)
Logs turn the index laws inside out: log(ab) = log a + log b is Law 1 read backwards. Every slide rule ever built ran on it.
Worked examples
Simplify 4x³ × 3x⁵
Simplify 15a⁷ ÷ 3a²
Simplify (2y⁴)³
Exam-style (GCSE): evaluate 27−2/3
Quick check
What you've learned
A power counts copies of the base. Multiplying powers of the same base puts the copies together, so indices add. Dividing cancels copies in pairs, so indices subtract. A power of a power is groups of groups, so indices multiply. Pushing Law 2 past zero gives a⁰ = 1 and a⁻ⁿ = 1/aⁿ; pushing Law 1 into fractions gives a1/n as the nth root. Coefficients are handled with ordinary arithmetic, and each letter is counted separately. When in doubt, draw the blocks.