Integers, Powers & Roots — GCSE Mathematics Revision
Revise Integers, Powers & Roots for GCSE Mathematics. Step-by-step explanation, worked examples, common mistakes and exam-style practice aligned to AQA, Edexcel and OCR.
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Go to Factors, Multiples & PrimesWhat is Integers, Powers & Roots?
Integers are whole numbers including negatives and zero. Powers (or indices) tell you how many times to multiply a number by itself — for example, 2³ = 2 × 2 × 2 = 8. Roots are the inverse of powers: the square root of 9 is 3 because 3² = 9. You need to be confident with prime factorisation, index laws, and recognising cube numbers up to 10³.
Board notes: All boards (AQA, Edexcel, OCR) test index laws at both Foundation and Higher. Negative and fractional indices are Higher only.
Step-by-step explanationWorked example
Simplify 2⁴ × 2³ ÷ 2⁵. Using index laws: add powers when multiplying (4+3=7), subtract when dividing (7-5=2). Answer: 2² = 4.
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Common mistakes
- 1Confusing (-3)² = 9 with -3² = -9 — the brackets matter because the exponent only applies to what is directly before it.
- 2Forgetting that any number to the power of 0 equals 1, not 0.
- 3Mixing up square roots and cube roots — √64 = 8 but ∛64 = 4.
- 4Not simplifying index expressions fully, e.g. leaving 2⁴ × 2³ as two separate terms instead of 2⁷.
Integers, Powers & Roots exam questions
Exam-style questions for Integers, Powers & Roots with mark-scheme style solutions and timing practice. Aligned to AQA, Edexcel and OCR specifications.
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Step-by-step method
Step-by-step explanation
4 steps · Worked method for Integers, Powers & Roots
Core concept
Integers are whole numbers including negatives and zero. Powers (or indices) tell you how many times to multiply a number by itself — for example, 2³ = 2 × 2 × 2 = 8. Roots are the inverse of powers: …
Frequently asked questions
What are the index laws for GCSE Maths?
The three main index laws are: aᵐ × aⁿ = aᵐ⁺ⁿ (multiply → add powers), aᵐ ÷ aⁿ = aᵐ⁻ⁿ (divide → subtract powers), and (aᵐ)ⁿ = aᵐⁿ (power of a power → multiply). You also need a⁰ = 1 and a⁻ⁿ = 1/aⁿ.
Do I need to know cube numbers for GCSE?
Yes. You should memorise cube numbers up to 10³ = 1000 and their cube roots. These appear in non-calculator papers and in questions on volume.