Indices and Powers Explained — KS3 Maths

An index (plural: indices) tells you how many times a number — called the base — is multiplied by itself. Writing 2⁵ is shorthand for 2 × 2 × 2 × 2 × 2 = 32. The 5 is the index; the 2 is the base.

Key vocabulary

Term | Definition | Example
Base | The number being multiplied | 3 in 3⁴
Index (exponent / power) | How many times the base is multiplied by itself | 4 in 3⁴
Index notation | Shorthand using a raised number | 3⁴
Value | The result of evaluating | 3⁴ = 81

The three laws of indices at KS3

Law 1 — Multiplying powers (add the indices)

When multiplying two powers of the same base, add the indices:

aᵐ × aⁿ = aᵐ⁺ⁿ

Example: 5³ × 5⁴ = 5³⁺⁴ = 5⁷

Verify: 5³ = 125, 5⁴ = 625, 125 × 625 = 78 125. And 5⁷ = 78 125. ✓

Law 2 — Dividing powers (subtract the indices)

When dividing two powers of the same base, subtract the indices:

aᵐ ÷ aⁿ = aᵐ⁻ⁿ

Example: 6⁵ ÷ 6² = 6⁵⁻² = 6³ = 216

Law 3 — Power of a power (multiply the indices)

When raising a power to another power, multiply the indices:

(aᵐ)ⁿ = aᵐˣⁿ

Example: (2³)⁴ = 2³ˣ⁴ = 2¹² = 4096

Special indices

The zero index

Any non-zero number raised to the power of 0 equals 1:

a⁰ = 1 (for a ≠ 0)

Why? Using Law 2: a³ ÷ a³ = a³⁻³ = a⁰. But any number divided by itself equals 1. So a⁰ = 1.

Examples: 7⁰ = 1, 100⁰ = 1, (−4)⁰ = 1

Negative indices

A negative index means take the reciprocal:

a⁻ⁿ = 1 ÷ aⁿ = 1/aⁿ

Example: 2⁻³ = 1/2³ = 1/8

Example: 5⁻² = 1/5² = 1/25

Worked examples

Example 1 — evaluate using index notation

Work out 4³.

4³ = 4 × 4 × 4 = 16 × 4 = 64

Example 2 — apply Law 1

Simplify 3² × 3⁵, leaving your answer in index form.

3² × 3⁵ = 3²⁺⁵ = 3⁷

(No need to evaluate; the question says "index form".)

Example 3 — apply Law 2

Simplify 7⁶ ÷ 7², leaving your answer in index form.

7⁶ ÷ 7² = 7⁶⁻² = 7⁴

Example 4 — apply Law 3

Simplify (x⁴)³.

(x⁴)³ = x⁴ˣ³ = x¹²

Example 5 — negative index

Write 3⁻⁴ as a fraction.

3⁻⁴ = 1/3⁴ = 1/81

Example 6 — mixed laws

Simplify (2³ × 2⁵) ÷ 2⁴.

Numerator: 2³ × 2⁵ = 2⁸

Then: 2⁸ ÷ 2⁴ = 2⁸⁻⁴ = 2⁴ = 16

Common mistakes

Mistake | Incorrect | Correct
Multiplying the base instead of adding the index | 2³ × 2⁴ = 4⁷ | 2³ × 2⁴ = 2⁷
Applying law to different bases | 2³ × 3⁴ = 6⁷ | Cannot simplify — different bases
Treating a⁰ as 0 | 5⁰ = 0 | 5⁰ = 1
Sign error with negative index | 2⁻³ = −8 | 2⁻³ = 1/8

Indices in the national curriculum

The DfE's KS3 mathematics programme of study requires pupils to use and interpret notation including index notation and to apply the laws of indices. The statutory national curriculum for Key Stage 3 and 4 confirms these rules are tested throughout Year 8 and Year 9 and form essential groundwork for GCSE algebra and number topics such as standard form and surds.

Frequently asked questions

What is the difference between an index, a power, and an exponent?

All three terms mean the same thing in KS3 maths: the small raised number that tells you how many times the base is multiplied by itself. In the expression 5³, the 3 is the index, power, or exponent — whichever term your teacher uses. "Indices" is simply the plural of "index".

When can I not use the multiplication law of indices?

Law 1 (aᵐ × aⁿ = aᵐ⁺ⁿ) only works when the base is the same. You cannot simplify 3² × 5⁴ using this law because 3 and 5 are different bases. In that case, you must evaluate each power separately and then multiply: 9 × 625 = 5625.

What does a power of 1 mean?

Any number to the power of 1 equals itself: a¹ = a. This fits the pattern — 1 tells you the base appears once in the multiplication. So 8¹ = 8, 100¹ = 100.

Can indices be fractions?

Yes — fractional indices are introduced at GCSE. A fractional index represents a root: a^(1/2) = √a, and a^(1/3) = ∛a. At KS3, you do not need to use fractional indices, but recognising that √a = a^(1/2) is useful background knowledge.

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