Squares, Cubes and Roots

What you will learn

• What square numbers and square roots mean.
• What cube numbers and cube roots mean.
• How to use squared and cubed notation.
• Common square and cube numbers to recognise.
• How to check roots and avoid sign mistakes.
• How roots appear in area, volume and checking questions.

Why this matters

Squares and roots appear in number questions, area, Pythagoras, graphs, surds and algebra. Cubes and cube roots appear in volume, powers and calculator questions.

Knowing the common values helps you work faster and makes later topics feel much less mysterious.

Prior knowledge

You should already be comfortable with:

• times tables and repeated multiplication,
• prime factorisation,
• positive and negative numbers,
• using a calculator carefully when it is allowed.

Clear explanation

To square a number, multiply it by itself. The notation 5² means 5 × 5.

A square root reverses squaring. It asks: what number was squared to make this?

Cube numbers

To cube a number, multiply it by itself three times. The notation 3³ means 3 × 3 × 3.

A cube root reverses cubing.

Roots undo powers

A useful habit is to check roots by multiplying back. If √81 = 9, then 9 × 9 should give 81.

Common values worth knowing

You do not need to memorise every possible power, but the common GCSE values make non-calculator questions much faster. A short recall list also helps you spot when a calculator answer is unreasonable.

Negative signs and brackets

Brackets matter when negative numbers are squared. (−4)² means (−4) × (−4), so the answer is 16. But −4² is usually read as −(4²), so the answer is −16.

In geometry, square roots often find a side length from an area, and cube roots often find an edge length from a volume.

Worked examples

Quick checks

Choose an answer, then check your thinking.

Practice questions

Answers and marking guidance

The practice answers are hidden under each question so you can try first. For this topic, marks often come from using the correct inverse operation, showing the multiplication that checks a root, recalling common square and cube values accurately, keeping square roots and cube roots separate, including units in area or volume contexts, and treating brackets around negative numbers carefully. When an area or volume is decimal or large, check by multiplying the proposed side or edge back.

Common mistakes

• Multiplying by 2 instead of squaring: 7² means 7 × 7, not 7 × 2.
• Multiplying by 3 instead of cubing: 4³ means 4 × 4 × 4, not 4 × 3.
• Forgetting that roots reverse powers: check √64 by asking which number squared gives 64.
• Mixing square roots and cube roots: √27 is not 3, but ∛27 is 3.
• Negative number traps: (-3)² = 9, but -3² is usually read as -(3²) = -9 unless brackets are shown.

Extension challenge

Find a number that is both a square number and a cube number. Explain why it works.

Exam-board guidance

Squares, cubes and roots appear across GCSE Maths specifications. The core skill is broadly the same, but exam boards may vary the style of question, calculator access, and how much the topic is linked to indices, geometry or problem-solving.

Next suggested lesson

Powers and Indices is a useful next step because squares and cubes are the first examples of index notation.

Back to GCSE Maths lessons