What you'll learn
The laws of indices are the rules for working with powers, and they make simplifying expressions and calculations far quicker. In this guide you will learn the basic index laws for multiplying, dividing and raising powers to powers, the meaning of zero, negative and fractional indices, and how to apply these rules to numbers and algebra. Indices underpin standard form, surds and much of algebra.
Key terms and definitions
Index (plural indices) — the power a number or letter is raised to (the small raised number).
Base — the number or letter being raised to a power.
Reciprocal — one divided by a number; a⁻¹ = 1/a.
Root — a fractional index represents a root (a^(1/2) = √a).
Power of a power — raising an already-indexed term to another power.
Core concepts
Multiplying and dividing powers
When multiplying powers of the same base, add the indices: aᵐ × aⁿ = aᵐ⁺ⁿ. For example, x³ × x⁴ = x⁷. When dividing, subtract the indices: aᵐ ÷ aⁿ = aᵐ⁻ⁿ. For example, x⁵ ÷ x² = x³. These only work when the bases are the same.
Power of a power
To raise a power to another power, multiply the indices: (aᵐ)ⁿ = aᵐⁿ. For example, (x²)³ = x⁶. Remember to apply a power to everything inside a bracket: (3x²)³ = 27x⁶ (the 3 is also cubed).
Zero index
Any non-zero number to the power zero equals 1: a⁰ = 1. This follows from the division rule (aⁿ ÷ aⁿ = a⁰ = 1).
Negative indices
A negative index means the reciprocal: a⁻ⁿ = 1/aⁿ. For example, 2⁻³ = 1/2³ = 1/8, and x⁻² = 1/x². To remove a negative power, take the reciprocal of the base.
Fractional indices
A fractional index represents a root: a^(1/n) = ⁿ√a. So a^(1/2) = √a and a^(1/3) = ∛a. A fraction like a^(m/n) means (ⁿ√a)ᵐ — take the root, then raise to the power: 8^(2/3) = (∛8)² = 2² = 4.
Worked examples
Example 1: Multiplying and dividing
Simplify x⁶ × x² ÷ x³.
Add then subtract indices: 6 + 2 − 3 = 5, so the answer is x⁵.
Example 2: Negative index
Evaluate 5⁻².
5⁻² = 1/5² = 1/25.
Example 3: Fractional index
Evaluate 16^(3/4).
16^(1/4) = 2 (since 2⁴ = 16), then 2³ = 8.
Common mistakes and how to avoid them
Multiplying indices when you should add. Add when multiplying powers; multiply only for a power of a power.
Forgetting to apply a power to coefficients. (2x³)² = 4x⁶ — the 2 is also squared.
Misreading negative indices. a⁻ⁿ = 1/aⁿ, not a negative number.
Mixing up the root and the power in fractions. In a^(m/n), n is the root and m is the power.
Using the rules with different bases. The add/subtract rules only work when the bases are the same.
Exam technique for Laws of Indices
Identify the operation — multiplying (add), dividing (subtract), power of power (multiply).
Deal with coefficients separately from the index laws, then combine.
Rewrite negative indices as reciprocals and fractional indices as roots.
Work fractional powers in steps — root first, then power (or vice versa).
Check the base is the same before adding or subtracting indices.
Quick revision summary
The laws of indices govern powers of the same base: multiply → add indices (aᵐ × aⁿ = aᵐ⁺ⁿ), divide → subtract indices (aᵐ ÷ aⁿ = aᵐ⁻ⁿ), and power of a power → multiply indices ((aᵐ)ⁿ = aᵐⁿ), remembering to apply the power to any coefficient too. A zero index gives 1 (a⁰ = 1). A negative index means the reciprocal (a⁻ⁿ = 1/aⁿ). A fractional index means a root: a^(1/n) = ⁿ√a, and a^(m/n) = (ⁿ√a)ᵐ, so you take the root then raise to the power (16^(3/4) = 2³ = 8). The common errors are multiplying indices when you should add, forgetting to raise coefficients, and misreading negative or fractional powers. Identify the operation, handle coefficients separately, rewrite negative and fractional indices, and always check the bases match. Mastering these rules speeds up algebra, standard form and surds alike.