Indices & Standard Form - AQA GCSE Mathematics Revision

Topic overview

What you need to know

Indices & Standard Form is a key part of GCSE Mathematics. Build fluency with the method, then apply it to unfamiliar and multi-step problems. A small number written above and to the right of a base number indicating how many times to multiply it by itself. E.g. 2³ = 2 × 2 × 2 = 8. aᵐ × aⁿ = aᵐ⁺ⁿ. When multiplying with the same base, add the powers. aᵐ ÷ aⁿ = aᵐ⁻ⁿ. When dividing with the same base, subtract the powers.

Exam tip

Write every stage of your method. Method marks can still be earned when the final answer is wrong.

Revision notes

Core ideas

Secure these ideas first. Say each definition in your own words, then connect it to the topic overview.

Index / Power: A small number written above and to the right of a base number indicating how many times to multiply it by itself. E.g. 2³ = 2 × 2 × 2 = 8.
Law of Indices: Multiplication: aᵐ × aⁿ = aᵐ⁺ⁿ. When multiplying with the same base, add the powers.
Law of Indices: Division: aᵐ ÷ aⁿ = aᵐ⁻ⁿ. When dividing with the same base, subtract the powers.
Negative Index: a⁻ⁿ = 1/aⁿ. A negative power means the reciprocal.

Revision notes

Apply it in the exam

The exam will rarely ask for an isolated definition. Practise selecting the right idea and using it as part of a complete explanation or method.

Zero Index: a⁰ = 1 for any non-zero value of a. Any number to the power of zero equals 1.
Fractional Index: a^(1/n) = ⁿ√a and a^(m/n) = (ⁿ√a)ᵐ. A fractional power means a root.
Standard Form: A number written as A × 10ⁿ where 1 ≤ A < 10 and n is an integer. E.g. 4500 = 4.5 × 10³.
Standard Form (Small Numbers): Small numbers use negative powers of 10. E.g. 0.003 = 3 × 10⁻³.

Core knowledge

Key facts for Indices & Standard Form

8 essentials

Index / Power

A small number written above and to the right of a base number indicating how many times to multiply it by itself. E.g. 2³ = 2 × 2 × 2 = 8.

Law of Indices: Multiplication

aᵐ × aⁿ = aᵐ⁺ⁿ. When multiplying with the same base, add the powers.

Law of Indices: Division

aᵐ ÷ aⁿ = aᵐ⁻ⁿ. When dividing with the same base, subtract the powers.

Negative Index

a⁻ⁿ = 1/aⁿ. A negative power means the reciprocal.

Zero Index

a⁰ = 1 for any non-zero value of a. Any number to the power of zero equals 1.

Fractional Index

a^(1/n) = ⁿ√a and a^(m/n) = (ⁿ√a)ᵐ. A fractional power means a root.

Standard Form

A number written as A × 10ⁿ where 1 ≤ A < 10 and n is an integer. E.g. 4500 = 4.5 × 10³.

Standard Form (Small Numbers)

Small numbers use negative powers of 10. E.g. 0.003 = 3 × 10⁻³.

Active recall

Close the notes and answer these

1.Without looking, explain index / power and give one example or consequence.
2.Without looking, explain law of indices: multiplication and give one example or consequence.
3.Without looking, explain law of indices: division and give one example or consequence.
4.Without looking, explain negative index and give one example or consequence.

Content reviewed 23 July 2026 against the current linked specification.

Finished learning?

Mark the notes complete, then test what you can recall.

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