Knowledge Organiser: Powers and Roots
Part of Powers & Roots · GCSE GCSE Mathematics revision
This topic summary covers Knowledge Organiser: Powers and Roots within Powers & Roots for GCSE Mathematics. Revise Powers & Roots in Number for GCSE Mathematics with 15 exam-style questions and 22 flashcards. This is a high-frequency topic, so it is worth revising until the explanation feels precise and repeatable. It is section 14 of 14 in this topic. Use this topic summary to connect the idea to the wider topic before moving on to questions and flashcards.
Topic position
Section 14 of 14
Practice
15 questions
Recall
22 flashcards
Knowledge Organiser: Powers and Roots
Key Terms
- Power/index: How many times to multiply a number by itself
- Base: The number being multiplied (e.g. in 5³, the base is 5)
- Square number: Result of raising a number to the power 2
- Cube number: Result of raising a number to the power 3
- Square root (√): The number that, when squared, gives the original
- Cube root (∛): The number that, when cubed, gives the original
Must-Know Facts
- Squares to 15²: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225
- Cubes to 5³: 1, 8, 27, 64, 125
- (−3)² = 9 but −3² = −9 (brackets matter)
- √ symbol means positive root only (√9 = 3, not ±3)
- 2⁰ = 1; any non-zero number to the power 0 equals 1
- 2³ = 8 (not 6) — powers mean multiply, not add
Key Formulas
- aⁿ = a × a × a × … (n times)
- √a = the number whose square is a
- ∛a = the number whose cube is a
- a^(1/2) = √a; a^(1/3) = ∛a (Higher tier)
- a^(−n) = 1/aⁿ (negative index = reciprocal)
Common Mistakes
- 2³ = 6: Wrong — 2³ = 2 × 2 × 2 = 8, not 2 × 3
- √(a + b) = √a + √b: False — √(9 + 16) = √25 = 5, NOT 3 + 4 = 7
- Negative index: 2⁻³ = −8 is wrong — 2⁻³ = 1/8 (reciprocal, not negative)
- Square root of a negative: At GCSE, √(−4) has no real answer
- Confusing √ with ÷ 2: √16 = 4 (not 8) — root is NOT the same as halving
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Practice Questions for Powers & Roots
What is the value of 2⁻³?
Explain why x⁻¹ = 1/x
Quick Recall Flashcards
15 questions on Powers & Roots — practise free
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