Sheet № 128 · Foundation + Higher · AQA · Edexcel · OCR
Square Numbers and Cube Numbers –
Square numbers and cube numbers are building blocks of GCSE Maths. Knowing them by heart speeds up work on indices, surds, area, volume, and Pythagoras' theorem across both Foundation and Higher tiers. This guide lists the key values to memorise, explains roots, and provides worked examples.
§Key definitions
Question:
Work out 13² and state whether 150 is a square number.
Answer:
13² = 169. No, 150 is not a square number.
Q1 (Foundation):
Write down the values of 9², 11², and 14².
Q2 (Foundation):
Find ∛125.
Q3 (Higher):
Find the smallest number that is both a perfect square and a perfect cube.
§Formulas to memorise
n² = n × n (square); √(n²) = n (square root)
n³ = n × n × n (cube); ∛(n³) = n (cube root)
Worked example
Work out 13² and state whether 150 is a square number.
Working:
⚠ Common mistakes
- ✗Confusing squaring with doubling. 5² = 25, not 10. Squaring means multiplying by itself, not by 2.
- ✗Forgetting that negative numbers also have squares. (−3)² = 9. However, the principal square root √9 = 3 (positive).
- ✗Mixing up square and cube roots. √ means square root (two equal factors) and ∛ means cube root (three equal factors). Read the symbol carefully.
✦ Exam tips
- →Memorise squares from 1² = 1 to 15² = 225 and cubes from 1³ = 1 to 10³ = 1000.
- →Recognising square and cube numbers helps with prime factorisation and simplifying surds.
- →On non-calculator papers, use known squares and cubes to estimate roots of nearby numbers.
- →The key squares to know: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225.
- →The key cubes to know: 1, 8, 27, 64, 125, 216, 343, 512, 729, 1000.