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.

What Are Square and Cube Numbers?

A square number is the result of multiplying a whole number by itself. For example, 4² = 4 × 4 = 16, so 16 is a square number. The square root reverses this: √16 = 4.

A cube number is the result of multiplying a whole number by itself three times. For example, 3³ = 3 × 3 × 3 = 27, so 27 is a cube number. The cube root reverses this: ∛27 = 3.

You should memorise the squares from 1² to 15² and the cubes from 1³ to 10³. These values come up repeatedly in non-calculator papers and save significant time.

Key Formulas

Step-by-Step Method

1. To find a square number, multiply the number by itself.
2. To find a cube number, multiply the number by itself twice (three factors in total).
3. To find a square root, ask "what number multiplied by itself gives this value?"
4. To find a cube root, ask "what number multiplied by itself three times gives this value?"

Worked Example 1 — Foundation Level

Question: Work out 13² and state whether 150 is a square number.

Working:

Step 1 — 13² = 13 × 13 = 169.

Step 2 — The squares near 150 are 12² = 144 and 13² = 169. Since 150 falls between these and is not equal to either, it is not a square number.

Answer: 13² = 169. No, 150 is not a square number.

Worked Example 2 — Higher Level

Question: Evaluate √(225) + ∛(64).

Working:

Step 1 — √225: test values. 15² = 225, so √225 = 15.

Step 2 — ∛64: 4³ = 64, so ∛64 = 4.

Step 3 — 15 + 4 = 19.

Answer: 19

Worked Example 3 — Exam Style

Question: Find the value of n such that n³ = 512.

Working:

Step 1 — Test cube numbers: 7³ = 343, 8³ = 512. So n = 8.

Answer: n = 8

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.

Practice Questions

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.

Practise square numbers and cube numbers questions with instant feedback — completely free on GCSEMathsAI.

Related Topics

• Powers and Roots
• Indices and Index Laws
• Estimating Square Roots

Summary

• Square numbers result from multiplying a number by itself: n² = n × n.
• Cube numbers result from multiplying a number by itself three times: n³ = n × n × n.
• Square roots and cube roots are the inverse operations.
• Memorise squares up to 15² = 225 and cubes up to 10³ = 1000.
• Squaring is not the same as doubling — this is one of the most common misconceptions.
• Recognising square and cube numbers helps in many other topics including surds and Pythagoras.