Indices and Index Laws – GCSE Maths Guide

Indices (also called powers or exponents) are a shorthand for repeated multiplication, and the index laws are the rules that let you simplify expressions involving them. This topic spans Foundation and Higher tiers: Foundation students need the basic laws for positive integer powers, while Higher students must also handle zero, negative, and fractional indices. These rules underpin algebra, standard form, and growth and decay problems, so getting them right pays dividends across the entire exam. This guide covers every law with clear explanations, worked examples at both levels, and practice questions. For a full topic map, see our complete GCSE Maths topics list.

What Are the Index Laws?

The index laws are a set of rules for simplifying expressions that have the same base. Here they are in full:

Multiplication Law

a^m × a^n = a^(m+n) — when multiplying, add the powers

Division Law

a^m ÷ a^n = a^(m−n) — when dividing, subtract the powers

Power of a Power Law

(a^m)^n = a^(mn) — when raising a power to another power, multiply the powers

Zero Index

a^0 = 1 — any non-zero number to the power of zero equals 1

Negative Index (Higher)

a^(−n) = 1/a^n — a negative power means the reciprocal

Fractional Index (Higher)

a^(1/n) = ⁿ√a — the nth root of a

a^(m/n) = (ⁿ√a)^m = ⁿ√(a^m) — root first, then power (or vice versa)

Important Notes

These laws only work when the bases are the same. You cannot use them to simplify 2³ × 3² because the bases (2 and 3) are different.
The base can be a number or a variable (like x).

Step-by-Step Method

Applying the Multiplication Law

Check that the bases are the same.
Keep the base and add the indices: e.g. x⁴ × x³ = x⁷.

Applying the Division Law

Check that the bases are the same.
Keep the base and subtract the indices: e.g. y⁸ ÷ y² = y⁶.

Applying the Power of a Power Law

Keep the base.
Multiply the indices: e.g. (z³)⁴ = z¹².

Evaluating Negative Indices

Take the reciprocal of the base.
Apply the positive version of the power: e.g. 5⁻² = 1/5² = 1/25.

Evaluating Fractional Indices

The denominator of the fraction tells you which root to take.
The numerator tells you the power to raise it to.
It is usually easier to take the root first, then apply the power: e.g. 8^(2/3) = (∛8)² = 2² = 4.

Worked Example 1 — Foundation Level

Question: Simplify 3x⁴y² × 5x³y.

Working:

Step 1 — Multiply the coefficients: 3 × 5 = 15.

Step 2 — Apply the multiplication law to x: x⁴ × x³ = x⁷.

Step 3 — Apply the multiplication law to y: y² × y¹ = y³.

Answer: 15x⁷y³

Worked Example 2 — Higher Level

Question: Evaluate 27^(−2/3).

Working:

Step 1 — Deal with the negative index: 27^(−2/3) = 1 / 27^(2/3).

Step 2 — Evaluate 27^(2/3). The denominator of 2/3 is 3, so take the cube root first: ∛27 = 3.

Step 3 — The numerator of 2/3 is 2, so square the result: 3² = 9.

Step 4 — Apply the reciprocal: 1/9.

Answer: 1/9

Common Mistakes

Adding indices when you should multiply (and vice versa). Remember: × means add the powers, (a^m)^n means multiply the powers. These are different operations.
Applying index laws to different bases. 2³ × 3² cannot be simplified using index laws — the bases must match.
Thinking a^0 = 0. Any non-zero number to the power of zero is 1, not 0. This catches out many students.
Mishandling negative indices. 2⁻³ = 1/8, not −8. A negative index means reciprocal, not a negative number.
Taking the power before the root with fractional indices. While mathematically you can do either order, taking the root first keeps the numbers small. 64^(2/3): cube root of 64 is 4, then 4² = 16. If you cube first: 64² = 4096, then ∛4096 — much harder without a calculator.

Exam Tips

Write out each law you are using. When simplifying expressions, annotate your working: "multiplication law: add powers". This helps you avoid errors and can earn method marks.
Fractional indices almost always appear on non-calculator Higher papers. Practise evaluating values like 16^(3/4), 125^(2/3), and 32^(−3/5) until they are automatic. See our formulas guide for the key index law rules.
Look out for hidden index law questions in algebra. Simplifying algebraic fractions, solving equations, and working with surds all draw on these laws.
If stuck, rewrite everything using the definition. For example, x⁻¹ = 1/x and x^(1/2) = √x. This can make unfamiliar problems feel manageable.

Practice Questions

Q1 (Foundation): Simplify p⁵ × p³ ÷ p².

Answer: Add then subtract powers: p^(5+3−2) = p⁶.

Q2 (Higher): Evaluate 16^(3/4).

Answer: ⁴√16 = 2. Then 2³ = 8. So 16^(3/4) = 8.

Q3 (Higher): Simplify (2x³)⁴.

Answer: Apply the power to everything inside the bracket: 2⁴ × (x³)⁴ = 16x¹².

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Summary

Index laws apply only when the bases are the same.
Multiplication: add the powers. Division: subtract the powers. Power of a power: multiply the powers.
Any non-zero value raised to the power 0 equals 1.
A negative index means take the reciprocal.
A fractional index means take a root (denominator) and raise to a power (numerator).
Take the root first when evaluating fractional indices to keep numbers manageable.
Index laws are used throughout algebra, standard form, and growth/decay problems.

Indices and Index Laws –