GCSE & IGCSE maths practice

Laws of indices questions and answers

The laws of indices work for powers of the same base: multiply → add the powers ( $a^m \times a^n = a^{m+n}$ ); divide → subtract ( $a^m \div a^n = a^{m-n}$ ); power of a power → multiply ( $(a^m)^n = a^{mn}$ ). Also $a^0 = 1$ , $a^{-n} = \dfrac{1}{a^n}$ and (Higher) $a^{1/n} = \sqrt[n]{a}$ .

8 questions · Foundation and Higher tier · Every answer is checked as you type it, with a full worked solution. Updated 27 Oct 2026.

Start the questions Laws of indices: lesson Laws of indices guide

Where laws of indices comes up

GCSE (AQA, Edexcel, OCR)	Foundation and Higher: calculate with roots and integer indices. Higher only: calculate with fractional indices.
Cambridge IGCSE (0580)	Core: understand and use indices (positive, zero and negative integers) and the rules of indices. Extended: fractional indices.

The method

Multiply: add the indices. Divide: subtract.
Power of a power: multiply the indices.
a⁰ = 1; a⁻ⁿ = 1 ÷ aⁿ.
Fractional (Higher): a^(m/n) = (ⁿ√a)ᵐ: root first, then power.

Laws of indices questions

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Warm-up

Question 1Warm-upFoundation

$3^2 \times 3^4 = 3^n$ . Find $n$ .

Question 2Warm-upFoundation

$5^7 \div 5^3 = 5^n$ . Find $n$ .

Exam-style

Question 3Exam-styleFoundation

$(2^3)^4 = 2^n$ . Find $n$ .

Question 4Exam-styleFoundation

Work out $4^0$ .

Question 5Exam-styleFoundation

Work out $2^{-3}$ as a fraction or decimal.

Question 6Exam-styleFoundation

Simplify $x^5 \times x^3$ .

Challenge

Question 7ChallengeHigher

Work out $27^{2/3}$ .

Question 8ChallengeHigher

Work out $16^{-1/2}$ as a fraction.

Common mistakes

Multiplying the indices when multiplying powers (3² × 3⁴ is 3⁶, not 3⁸).
Thinking a⁰ = 0.
Making a negative index give a negative answer.

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Sources

The questions are our own. The “where it comes up” notes are from each board's published specification, checked on 26 October 2026:

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