Indices, Powers & Roots — Year 8 Revision Notes
These notes cover everything you need on indices for Year 8 (Stage 8) of Cambridge Lower Secondary Maths: writing powers, finding roots, and using the laws of indices to simplify calculations.
Index notation, squares and cubes
A power is written as a base with a small raised index, so 2⁵ has base 2 and index 5 and means 2 × 2 × 2 × 2 × 2 = 32. Squaring means raising to the power 2 (6² = 36) and cubing means raising to the power 3 (4³ = 64). Learn the square numbers up to 15² and the cube numbers up to 5³ by heart — they appear constantly.
Key Facts & Formulas
- aⁿ = a × a × … (n times)
- 6² = 36
- 4³ = 64
Tips
- Read 5³ as 'five cubed', not 'five times three'.
- Memorise square numbers to 15² and cubes to 5³ to save time.
Square roots and cube roots
A square root reverses squaring: √49 = 7 because 7² = 49. A cube root reverses cubing: ∛27 = 3 because 3³ = 27. If a number is not a perfect square, estimate its root by finding the two whole numbers it lies between — √20 is between √16 = 4 and √25 = 5, so it is about 4.5.
Key Facts & Formulas
- √49 = 7
- ∛27 = 3
Tips
- Check a root by squaring or cubing your answer.
- For a non-exact root, name the two perfect squares it sits between.
The laws of indices
When you multiply powers of the same base you add the indices, and when you divide them you subtract: 2³ × 2² = 2⁵ and 2⁶ ÷ 2² = 2⁴. Any non-zero number to the power zero is 1. These laws only work when the bases are the same.
Key Facts & Formulas
- aᵐ × aⁿ = aᵐ⁺ⁿ
- aᵐ ÷ aⁿ = aᵐ⁻ⁿ
- a⁰ = 1
Tips
- Check the bases match before adding or subtracting indices.
- Add indices for ×, subtract for ÷ — never multiply the bases.
Revision Checklist
- I can write and evaluate powers using index notation
- I know the square numbers to 15² and the cube numbers to 5³
- I can find square and cube roots and estimate non-exact roots
- I can use the laws of indices to multiply and divide powers
Frequently Asked Questions
Do the laws of indices work for different bases?
No. aᵐ × aⁿ = aᵐ⁺ⁿ only works when the base is the same. 2³ × 5² cannot be combined into a single power because the bases (2 and 5) are different.
Build strong foundations in Indices, Powers & Roots
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