Square Numbers and Square Roots
Master square numbers and square roots for GCSE Maths with structured, exam-style practice. This Foundation resource covers square and cube numbers and their roots and includes a complete mark scheme showing the steps examiners reward. Suitable for AQA, Edexcel and OCR. Free to download as a PDF. Square numbers come from multiplying an integer by itself: 1, 4, 9, 16, 25.
Free downloads
These worksheets and mark schemes are original, written for Virtus Academy and checked against the current AQA, Edexcel and OCR specifications. Every worksheet comes with a full mark scheme.
Challenge / Extension
Stretch yourself beyond the basics.
Topic overview
A square number is the result of multiplying a whole number by itself. The first few are \(1, 4, 9, 16, 25, 36, 49, 64, 81\) and \(100\), and they are worth knowing up to at least \(15^2 = 225\).
The square root reverses the operation. Since \(7 \times 7 = 49\), the square root of \(49\) is \(7\). Every positive number has both a positive and a negative square root, though questions usually want the positive one unless they say otherwise.
Squares and roots appear throughout the GCSE, from area and Pythagoras to quadratic equations. Recognising a square number instantly often turns a hard-looking question into an easy one.
Revision notes
Square numbers
A square number is \(n \times n\) for a whole number \(n\), written \(n^2\).
Learn them to \(15^2\): \(1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225\). They are called squares because that many dots form a perfect square.
Square roots
The square root asks which number multiplied by itself gives this value, written \(\sqrt{\ }\).
So \(\sqrt{81} = 9\). Strictly \(-9\) also squares to \(81\), so a quadratic gives both roots, but \(\sqrt{81}\) on its own means the positive one.
Estimating roots that are not exact
If a number lies between two squares, its root lies between the two roots.
Since \(\sqrt{49} = 7\) and \(\sqrt{64} = 8\), the square root of \(60\) lies between \(7\) and \(8\), and closer to \(8\).
Key points
- A square number is a whole number multiplied by itself.
- Learn the squares up to \(15^2 = 225\).
- The square root reverses squaring.
- Every positive number has a positive and a negative square root.
- \(\sqrt{\ }\) on its own means the positive root.
- Estimate a non-exact root by finding the squares either side.
Worked examples
Example 1
Work out \(13^2\).
Working
Example 2
Work out \(\sqrt{144}\).
Working
Example 3
Estimate \(\sqrt{50}\) between two consecutive whole numbers.
Working
Common mistakes
Doubling instead of squaring.
6² is 36, not 12. Squaring means multiplying by itself, not by 2.
Thinking the square root of a number is always smaller.
The square root of 0.25 is 0.5, which is larger. This holds only for numbers above 1.
Forgetting the negative root in a quadratic.
If x² = 25 then x = 5 or x = −5. Giving only one root loses a mark.
Confusing squaring with square rooting.
Read the question carefully — the two operations undo each other.
Exam tips
- Memorise the squares to 15² so you recognise them instantly.
- For non-exact roots, name the two whole numbers the answer sits between.
- Give both roots when solving an equation of the form x² = k.
- Check by squaring your answer back.
Key terms
- Square number
- The result of multiplying a whole number by itself.
- Square root
- The number that multiplies by itself to give a given value.
- Perfect square
- A number whose square root is a whole number.
- Consecutive
- Following one after another, such as 7 and 8.
Related topics
Written and reviewed against the current AQA, Edexcel and OCR specifications. Spotted an error? Let us know.