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This lesson covers the fundamental laws of indices (also called exponents or powers) that underpin all work with exponentials and logarithms in the Edexcel 9MA0 A-Level Mathematics specification. You must be fluent with these rules before tackling exponential equations and logarithmic manipulation.
An index (plural: indices) tells you how many times a base number is multiplied by itself. For example:
In general, aⁿ means "a multiplied by itself n times," where a is the base and n is the index (or exponent).
When you multiply two powers with the same base, you add the indices.
Example: 3⁴ × 3² = 3⁴⁺² = 3⁶ = 729
Why it works: 3⁴ × 3² = (3 × 3 × 3 × 3) × (3 × 3) = 3⁶
When you divide two powers with the same base, you subtract the indices.
Example: 5⁷ ÷ 5³ = 5⁷⁻³ = 5⁴ = 625
When you raise a power to another power, you multiply the indices.
Example: (2³)⁴ = 2³ˣ⁴ = 2¹² = 4096
Any non-zero number raised to the power zero equals 1.
Why it works: Using Law 2, aⁿ ÷ aⁿ = aⁿ⁻ⁿ = a⁰. But aⁿ ÷ aⁿ = 1. Therefore a⁰ = 1.
A negative index means "one over" the positive power.
Example: 4⁻² = 1/4² = 1/16
Why it works: Using Law 2, a⁰ ÷ aⁿ = a⁰⁻ⁿ = a⁻ⁿ. But a⁰ ÷ aⁿ = 1/aⁿ. Therefore a⁻ⁿ = 1/aⁿ.
A fractional index with numerator 1 means the nth root.
Examples:
A fractional index m/n means "take the nth root, then raise to the power m" (or vice versa).
Example: 8^(2/3) = (³√8)² = 2² = 4
Exam Tip: When evaluating expressions like 27^(2/3), always take the root first to keep numbers manageable: ³√27 = 3, then 3² = 9. Do not compute 27² = 729 first.
In A-Level questions you will often need to apply several laws in sequence.
Example: Simplify (2x³y²)⁴ ÷ (4x²y)²
Step 1 — Expand brackets:
Step 2 — Divide:
Surds can be written using fractional indices:
This is essential for differentiation and integration later in the course.
Example: Write (4√x³) in index form.
| Mistake | Correction |
|---|---|
| (a + b)² = a² + b² | (a + b)² = a² + 2ab + b² — you cannot distribute a power over addition |
| a⁻² = -a² | a⁻² = 1/a² — a negative index is a reciprocal, not a negative number |
| a^(1/2) × a^(1/2) = a^(1/4) | a^(1/2) × a^(1/2) = a^(1/2 + 1/2) = a¹ — add the indices |
| (a²)³ = a⁵ | (a²)³ = a⁶ — multiply (not add) the indices |