Index laws: revision notes and practice

What you will learn

• Explain what a base and an index mean.
• Choose the correct law for multiplication, division or a power of a power.
• Interpret zero and negative indices without confusing them with negative numbers.
• Higher extension: connect fractional indices to roots.

Key rules

Multiply and divide

aᵐ × aⁿ = aᵐ⁺ⁿ. For a ≠ 0, aᵐ ÷ aⁿ = aᵐ⁻ⁿ. Keep the base; add or subtract the indices.

Power of a power

(aᵐ)ⁿ = aᵐⁿ. Multiply the indices, rather than adding them.

Zero and negative indices

For a ≠ 0, a⁰ = 1 and a⁻ⁿ = 1/aⁿ. A negative index means a reciprocal, not a negative answer.

Worked examples

Example 1: Simplify 4³ × 4² as a single power of 4.

1. The bases match, so add the indices: 3 + 2 = 5.
2. Keep the base as 4.

4⁵

Example 2: Simplify (3²)³ ÷ 3⁴ as a single power of 3.

1. First multiply the bracket indices: 2 × 3 = 6.
2. Then subtract the division index: 6 − 4 = 2.

3²

A practice question

Simplify 4⁵ × 4⁵. Give your answer as a single power of 4.

1. Read a power before calculating

In 3⁴, the base is 3 and the index, also called the exponent, is 4. It means 3 × 3 × 3 × 3, which is 81. It does not mean 3 × 4. Writing out a small power as repeated multiplication is a reliable way to understand what a rule is doing.

The letters in x⁵ mean exactly the same kind of repeated multiplication: x × x × x × x × x. An instruction to simplify as a single power usually wants an answer such as x⁷, rather than a decimal or a long product.

2. Multiply powers with the same base

When the bases match, count how many copies of that base occur altogether. For x³ × x⁴ there are three copies of x followed by another four, so the result is x⁷. Keep the base and add the indices. This is why multiplying powers uses addition in the exponent.

Deal with ordinary coefficients separately. For example, 2x³ × 5x² = 10x⁵: multiply 2 and 5, then combine the powers of x. The same-base rule does not directly combine 2³ × 3², because those bases are different.

aᵐ × aⁿ = aᵐ⁺ⁿ

3. Divide powers with the same base

Think of division as cancelling matching factors. In x⁶ ÷ x², two copies of x cancel from the numerator and denominator, leaving x⁴. Keep the base and subtract the denominator's index from the numerator's index. The order of the subtraction matters.

The base must be non-zero when division by one of its powers is involved. With a smaller power on top, such as x² ÷ x⁵, cancellation leaves 1/x³. The index law gives the equivalent expression x⁻³, so a negative exponent is a useful way to record a reciprocal.

aᵐ ÷ aⁿ = aᵐ⁻ⁿ, for a ≠ 0

4. Raise a power to another power

A bracket followed by a power repeats the entire bracket. For (x³)², you have x³ × x³, giving x⁶. Multiply the two indices. This is a different operation from adding indices when two powers are multiplied side by side.

If a coefficient is inside the bracket, it is raised to the outer power too. Thus (2x³)² = 2² × x⁶ = 4x⁶. Squaring only the x part would miss a factor of 2. Similarly, a power distributes across a product, but not across an addition: (x + 2)² is not x² + 4.

(aᵐ)ⁿ = aᵐⁿ

5. Understand the zero index

For a non-zero base, a⁰ equals 1. One explanation compares a³ ÷ a³: a number divided by itself is 1, while the division law gives a³⁻³ = a⁰. Both descriptions must agree.

A zero index does not make the answer zero. For example, 7⁰ = 1, so 4 × 7⁰ = 4. This rule is stated for a ≠ 0; do not use it to assign an ordinary value to 0⁰ in these GCSE calculations.

a⁰ = 1, for a ≠ 0

6. Understand negative indices

A negative index tells you to take the reciprocal of the corresponding positive power. For example, 5⁻² = 1/5² = 1/25. It does not mean −25. Keep this distinction between a negative exponent and a negative value clear.

Index laws still work with integer exponents: 5³ × 5⁻² = 5¹ = 5. You can check this as 125 × 1/25. When simplifying algebra, x⁻² may be written 1/x² if the instruction asks for positive indices, with x ≠ 0.

a⁻ⁿ = 1/aⁿ, for a ≠ 0

7. Protect signs and the order of operations

Brackets decide whether a negative sign belongs to the base. The expression (−3)² is (−3) × (−3) = 9. In −3², the power is calculated before the leading minus, giving −9. A calculator will also distinguish these expressions if you enter them correctly.

Simplify powers inside brackets first, then apply the remaining operations. In (2²)³ ÷ 2⁴, multiply the bracket indices to get 2⁶, then subtract 4 to obtain 2². Showing those intermediate powers helps you avoid applying the right law to the wrong operation.

8. Higher extension: fractional indices

For a positive base, an index of 1/n represents the nth root. Therefore 16¹ᐟ² = √16 = 4, and 27¹ᐟ³ = ∛27 = 3. The denominator tells you the root; it does not tell you to divide the base by that number.

A numerator other than 1 adds a power. For example, 27²ᐟ³ = (∛27)² = 3² = 9. A negative fractional index adds a reciprocal: 16⁻¹ᐟ² = 1/√16 = 1/4. Use positive bases in these examples so that real-root and sign conventions do not create ambiguity.

aᵐᐟⁿ = (ⁿ√a)ᵐ, for a > 0 and positive integer n

Check your understanding

How to earn the marks

• Identify the operation first: multiply, divide, or raise a power.
• Check that the bases match before adding or subtracting indices.
• Simplify coefficients and powers separately, and show one intermediate line.
• If positive indices are requested, rewrite negative powers as reciprocals.

A common mistake

Do not apply the same-base rule to 2³ × 3². The bases are different.

Course context and official specifications

This guide covers selected skills. Some extensions are Higher content; choose your official tier separately from practice difficulty in the app. These are original questions and explanations.

• AQA GCSE Mathematics — N6, N7, A4

  Integer powers and index laws are Foundation content; fractional powers and exact surds extend into Higher content. This practice bank does not cover every index skill.

• Edexcel GCSE Mathematics — N6, N7, A4

  Integer powers and index laws are Foundation content; fractional powers and exact surds extend into Higher content. This practice bank does not cover every index skill.

• OCR GCSE Mathematics — 3.01

  Integer powers and index laws are Foundation content; fractional powers and exact surds extend into Higher content. This practice bank does not cover every index skill.

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