Simplify \(\sqrt{16}\).
Which number squared gives 16?
\(\sqrt{16} = 4\).
Surds
Surds are irrational numbers written in exact form using square roots. They are closely linked to powers and roots and simplifying expressions, and in GCSE Maths you need to simplify surds and sometimes rationalise denominators without converting to decimals.
Overview
A surd is a root that cannot be written as a simple whole number or terminating decimal.
Instead of writing an approximate decimal, we keep the answer in exact form.
\( \sqrt{2}, \ \sqrt{3}, \ \sqrt{5} \)
For example, \( \sqrt{2} \) is not a whole number and does not end, so it is a surd.
But \( \sqrt{9} = 3 \), so that is not a surd.
What you should understand after this topic
- Understand what a surd is
- Simplify surds
- Add and subtract like surds
- Multiply surds
- Understand why exact form matters in maths
Key Definitions
Surd
An irrational root written in exact form.
Exact Value
An answer kept exactly, not rounded to a decimal.
Simplify
Rewrite the surd in a shorter or neater exact form.
Square Number Factor
A square number inside the root that can be taken out.
Like Surds
Surds with the same root part, such as \(3\sqrt{2}\) and \(5\sqrt{2}\).
Irrational Number
A number that cannot be written exactly as a fraction.
Key Rules
Perfect square roots are not surds
\( \sqrt{16} = 4 \)
Split square factors
\( \sqrt{ab} = \sqrt{a}\sqrt{b} \)
Only like surds combine
\( 2\sqrt{3} + 5\sqrt{3} = 7\sqrt{3} \)
Multiply numbers and roots separately
\( \sqrt{2} \times \sqrt{8} = \sqrt{16} = 4 \)
Quick Comparison
| Expression | Simplified Form |
|---|---|
| \( \sqrt{12} \) | \( 2\sqrt{3} \) |
| \( \sqrt{18} \) | \( 3\sqrt{2} \) |
| \( \sqrt{50} \) | \( 5\sqrt{2} \) |
| \( \sqrt{72} \) | \( 6\sqrt{2} \) |
How to Solve
Step 1: Understand surds
A surd is a root that cannot be written exactly as a whole number or fraction.
\( \sqrt{2} \approx 1.414\ldots \)
Because the decimal is not exact, we leave the answer as \(\sqrt{2}\).
Exam tip: A surd is an exact answer.
Step 2: Recognise surds
Not a surd
\( \sqrt{25} = 5 \)
Surd
\( \sqrt{7} \)
Not a surd
\( \sqrt{49} = 7 \)
Surd
\( \sqrt{11} \)
Step 3: Simplify surds
Look for the largest square number factor inside the root.
\( \sqrt{12} = \sqrt{4 \times 3} = 2\sqrt{3} \)
The square factor \(4\) comes out as \(2\).
Step 4: Add and subtract like surds
You can only combine surds if the root part is the same.
\( 2\sqrt{3} + 5\sqrt{3} = 7\sqrt{3} \)
\( 4\sqrt{2} - \sqrt{2} = 3\sqrt{2} \)
\(2\sqrt{2} + 3\sqrt{5}\) cannot be combined.
Step 5: Multiply surds
Multiply the numbers outside the roots and multiply the values inside the roots.
\( \sqrt{3} \times \sqrt{12} = \sqrt{36} = 6 \)
\( 2\sqrt{5} \times 3\sqrt{2} = 6\sqrt{10} \)
Step 6: Expand brackets with surds
\( \sqrt{2}(\sqrt{2} + 3) = 2 + 3\sqrt{2} \)
Multiply the outside term by each term inside the bracket.
Step 7: Keep answers exact
Surds are useful because they avoid rounding errors.
Do not change surds into decimals unless the question asks for a decimal.
See powers and roots for square root basics.
Step 8: Exam method summary
- Check whether the root is exact.
- Look for the largest square factor.
- Simplify the surd.
- Only combine like surds.
- Keep answers in exact form unless told otherwise.
Example Questions
Edexcel
Exam-style questions inspired by Edexcel GCSE Mathematics.
Edexcel
Simplify \( \sqrt{50} \).
Edexcel
Simplify \( \sqrt{72} \).
Edexcel
Express \( 3\sqrt{5} + 2\sqrt{5} \) in its simplest form.
Edexcel
Express \( 5\sqrt{3} - 2\sqrt{3} \) in its simplest form.
Edexcel
Find the value of \( \sqrt{9} \times \sqrt{5} \). Give your answer in surd form.
AQA
Exam-style questions based on the AQA GCSE Mathematics specification, focusing on accurate manipulation and simplification of surds.
AQA
Simplify \( \sqrt{18} + \sqrt{8} \).
AQA
Simplify \( \sqrt{12} \times \sqrt{3} \).
AQA
Expand and simplify \( \sqrt{2}(\sqrt{8} + \sqrt{18}) \).
AQA
Rationalise the denominator of \( \frac{1}{\sqrt{5}} \).
AQA
A student says that \( \sqrt{9} + \sqrt{16} = \sqrt{25} \).
Is the student correct?
Tick one box. Yes ☐ No ☐
Give a reason for your answer.
OCR
Exam-style questions aligned with OCR GCSE Mathematics, emphasising reasoning, algebraic manipulation, and rationalising denominators.
OCR
Simplify \( \sqrt{45} \).
OCR
Simplify \( 2\sqrt{3} \times 3\sqrt{6} \).
OCR
Simplify \( (\sqrt{5} + 2)(\sqrt{5} - 2) \).
OCR
Rationalise the denominator of \( \frac{3}{\sqrt{7}} \).
OCR
Show that \( \frac{1}{\sqrt{2} - 1} = \sqrt{2} + 1 \).
Exam Checklist
Step 1
Check whether the root is already a whole number.
Step 2
Look for the largest square number factor inside the surd.
Step 3
Only combine surds if the root parts are the same.
Step 4
After multiplying, simplify the final answer fully.
Most common exam mistakes
Simplifying error
Using a square factor, but not the largest one.
Adding error
Combining unlike surds such as \( \sqrt{2} + \sqrt{3} \).
Decimal error
Turning exact answers into decimals when the question wants surd form.
Final answer error
Stopping before the result is fully simplified.
Common Mistakes
These are common mistakes students make when working with surds in GCSE Maths.
Thinking every square root is a surd
Incorrect
A student treats all square roots as surds.
Correct
A surd is a root that cannot be simplified into an integer. For example, \(\sqrt{9} = 3\) is not a surd, but \(\sqrt{2}\) is.
Missing the largest square factor
Incorrect
A student simplifies using a smaller factor.
Correct
Always look for the largest square factor. For example, \(\sqrt{50} = \sqrt{25 \times 2} = 5\sqrt{2}\).
Adding unlike surds
Incorrect
A student adds surds with different radicands.
Correct
Only like surds can be combined. \(2\sqrt{3} + 5\sqrt{3} = 7\sqrt{3}\), but \(\sqrt{2} + \sqrt{3}\) cannot be simplified further.
Converting to decimals too early
Incorrect
A student changes surds into approximations during working.
Correct
Keep answers in surd form unless the question asks for a decimal approximation.
Not simplifying after multiplication
Incorrect
A student multiplies surds but leaves the result unsimplified.
Correct
After multiplying, simplify fully. For example, \(\sqrt{3} \times \sqrt{12} = \sqrt{36} = 6\).
Try It Yourself
Practise simplifying and rationalising expressions involving surds.
Foundation
Foundation Practice
Simplify basic surds and recognise square numbers.
Simplify \(\sqrt{25}\).
Perfect square.
\(\sqrt{25} = 5\).
Simplify \(\sqrt{18}\).
18 = 9 × 2.
\(\sqrt{18} = \sqrt{9×2} = 3\sqrt{2}\).
Simplify \(\sqrt{50}\).
50 = 25 × 2.
\(\sqrt{50} = 5\sqrt{2}\).
Simplify \(\sqrt{12}\).
12 = 4 × 3.
\(\sqrt{12} = 2\sqrt{3}\).
Simplify \(\sqrt{27}\).
27 = 9 × 3.
\(\sqrt{27} = 3\sqrt{3}\).
Simplify \(2\sqrt{3} + 3\sqrt{3}\).
Add like terms.
2√3 + 3√3 = 5√3.
Simplify \(4\sqrt{2} + \sqrt{2}\).
Add coefficients.
4√2 + √2 = 5√2.
A student says \(\sqrt{12} = \sqrt{6}\). What is wrong?
Break into factors.
\(\sqrt{12} = 2\sqrt{3}\), not √6.
Simplify \(3\sqrt{5} + 2\sqrt{5}\).
Same surd.
3√5 + 2√5 = 5√5.
Higher
Higher Practice
Multiply surds and rationalise denominators.
Find \(\sqrt{3} × \sqrt{5}\).
Multiply inside the root.
\(\sqrt{15}\).
Find \(\sqrt{2} × \sqrt{8}\).
Multiply inside first.
\(\sqrt{16} = 4\).
Simplify \((\sqrt{3})^2\).
Square cancels root.
3.
Simplify \((\sqrt{7})^2\).
Square root squared.
7.
Rationalise \(\frac{1}{\sqrt{2}}\).
Multiply top and bottom by √2.
\(\frac{\sqrt{2}}{2}\).
Rationalise \(\frac{3}{\sqrt{3}}\).
Multiply by √3.
\(\frac{3√3}{3} = √3\).
Simplify \((2\sqrt{3})(3\sqrt{2})\).
Multiply numbers and roots.
2 × 3 = 6 and √3 × √2 = √6.
Simplify \((\sqrt{5})(\sqrt{20})\).
Multiply inside.
\(\sqrt{100} = 10\).
A student says \(\sqrt{2} + \sqrt{3} = \sqrt{5}\). What is wrong?
Like terms only.
Different surds cannot be combined.
Rationalise \(\frac{2}{\sqrt{5}}\).
Multiply by √5.
\(\frac{2√5}{5}\).
Games
Practise this topic with interactive games.
Games coming soon.
Frequently Asked Questions
What is a surd?
A surd is an irrational number that cannot be simplified into an exact decimal.
How do I simplify surds?
Take out square factors from inside the root.
Why don’t we use decimals for surds?
Because decimals are approximate, while surds give exact answers.