What is a surd in Mathematics, and why is it important in the Edexcel IGCSE curriculum?

A surd is an irrational number that cannot be expressed as a simple fraction, often represented as a root that cannot be simplified to remove the root symbol. In the Edexcel IGCSE Mathematics curriculum, understanding surds is important because it helps students work with irrational numbers and perform operations like simplification, addition, subtraction, multiplication, and division involving surds.

How do you simplify a surd, and what are some examples?

To simplify a surd, you need to factor the number under the root into its prime factors and pair the factors to move them outside the root. For example, to simplify √50, you factor it as √(25×2), which simplifies to 5√2. Simplifying surds is a key skill in the Edexcel IGCSE Mathematics curriculum, as it helps in solving equations and simplifying expressions.

What are the rules for adding and subtracting surds?

To add or subtract surds, they must be like surds, meaning they have the same radicand (the number under the root). For example, 3√2 + 2√2 = 5√2. If the surds are not like, they cannot be directly added or subtracted. Understanding these rules is crucial for solving problems in the Numbers and the Number System section of the Edexcel IGCSE Mathematics syllabus.

How do you multiply and divide surds?

When multiplying surds, you multiply the numbers outside the roots and the numbers inside the roots separately. For example, √3 × √12 = √(3×12) = √36 = 6. For division, you divide the numbers inside the roots and simplify if possible. For example, √18 ÷ √2 = √(18/2) = √9 = 3. Mastery of these operations is essential for the Edexcel IGCSE Mathematics exam.

What is rationalizing the denominator, and how is it done with surds?

Rationalizing the denominator involves eliminating surds from the denominator of a fraction. This is done by multiplying the numerator and the denominator by a suitable surd. For example, to rationalize 1/√2, multiply by √2/√2 to get √2/2. This process is important in the Edexcel IGCSE Mathematics curriculum as it helps in simplifying expressions and solving equations involving surds.

Why is "Writing $\sqrt{a + b} = \sqrt{a} + \sqrt{b}$" a common mistake in Surds?

Why it happens: Looks like distribution. How to avoid it: √(a + b) √(a) + √(b). Test: √(9 + 16) = √(25) = 5, but √(9) + √(16) = 3 + 4 = 7. NOT the same.

Why is "Pulling out a small perfect square factor instead of the largest" a common mistake in Surds?

Why it happens: √(72) = √(4 × 18) = 2√(18) feels right but isn't simplified. How to avoid it: Find the LARGEST perfect square factor: 72 = 36 × 2 6√(2). If you accidentally do 4 × 18, simplify √(18) further. Source: 4MA1/1H — examiner reports 2022-2024

Why is "Adding $\sqrt{3} + \sqrt{5}$ as $\sqrt{8}$" a common mistake in Surds?

Why it happens: Confusing addition with multiplication. How to avoid it: Surds with DIFFERENT radicals can't be combined: √(3) + √(5) stays as is. Multiplication: √(3) × √(5) = √(15).

Why is "Multiplying ONLY the numerator by $\sqrt{c}$ (not the denominator)" a common mistake in Surds?

Why it happens: Hurried. How to avoid it: ALWAYS multiply BOTH parts by the same value. That's just '× 1' in disguise. Mark schemes deduct heavily for one-sided.

Why is "Using the same sign as the original instead of conjugate" a common mistake in Surds?

Why it happens: Forgetting to flip. How to avoid it: Conjugate FLIPS the sign on the surd. a + b√(c) a - b√(c). Then (a + b√(c))(a - b√(c)) = a^2 - b^2c — no surd. Source: 4MA1/1H Jan 2024 — examiner report Q19

Numbers and the Number SystemEdexcel IGCSE Mathematics (4MA1)

Surds

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Surds Study Notes — Edexcel IGCSE 4MA1 Higher Tier (2026 onwards)

Manipulate roots of non-perfect squares without losing precision. The conjugate trick is one of Edexcel's favourite A* tools — guaranteed on Paper 1H.

What you’ll learn

Mapped to the Cambridge IGCSE 4MA1 syllabus (2026 onwards).

1.6 — Use surds and simplify expressions involving surds.
1.6 — Rationalise the denominator of a fraction.
1.6 — Use surds in geometric and algebraic contexts.

Simplify surds

Pull out the LARGEST perfect-square factor.

Definition. A SURD is the root of a non-perfect square (or higher non-perfect power). $\sqrt{2}$ and $\sqrt[3]{5}$ are surds; $\sqrt{4} = 2$ is not.

Why surds matter. Many quantities (Pythagorean lengths, trig values, quadratic solutions) come out as surds. Edexcel mark schemes prefer EXACT answers in surd form rather than decimal approximations.

Method to simplify $\sqrt{n}$ :

Find the LARGEST perfect square factor of $n$ .
Use $\sqrt{ab} = \sqrt{a} \cdot \sqrt{b}$ .

Common perfect squares to know: $4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225$ .

Examples:

Surd	Largest perfect square factor	Simplified
$\sqrt{8}$	$4$	$\sqrt{4 \cdot 2} = 2\sqrt{2}$
$\sqrt{50}$	$25$	$5\sqrt{2}$
$\sqrt{72}$	$36$	$6\sqrt{2}$
$\sqrt{18}$	$9$	$3\sqrt{2}$
$\sqrt{45}$	$9$	$3\sqrt{5}$
$\sqrt{75}$	$25$	$5\sqrt{3}$
$\sqrt{200}$	$100$	$10\sqrt{2}$

Worked qualitative. Why pull out the LARGEST perfect square?

$\sqrt{72} = \sqrt{4 \cdot 18} = 2\sqrt{18}$ .
But $\sqrt{18}$ still simplifies: $2\sqrt{18} = 2 \cdot 3\sqrt{2} = 6\sqrt{2}$ .
Same answer, more steps. Largest factor → done in one step.

Edexcel tip. ALWAYS check: can the radical be simplified further? If $\sqrt{18}$ appears in your answer, you missed a factor. Pull out the $9$ .

Find LARGEST perfect square.
$\sqrt{ab} = \sqrt{a}\sqrt{b}$ .
Memorise common ones: $\sqrt{8}, \sqrt{18}, \sqrt{32}, \sqrt{50}, \sqrt{72}$ .
Check radical can't simplify further.

Add, subtract, multiply

Same radical: combine. Different: leave alone. FOIL works on surd brackets.

Addition and subtraction.

Surds with the SAME radical part combine like terms:

$5\sqrt{2} + 3\sqrt{2} = 8\sqrt{2}$ .
$7\sqrt{3} - 2\sqrt{3} = 5\sqrt{3}$ .

DIFFERENT radicals can't be combined:

$\sqrt{2} + \sqrt{3}$ stays as is.

Multiplication.

$\sqrt{a} \times \sqrt{b} = \sqrt{ab}$ .

Examples:

$\sqrt{2} \times \sqrt{3} = \sqrt{6}$ .
$\sqrt{8} \times \sqrt{2} = \sqrt{16} = 4$ .
$3\sqrt{2} \times 2\sqrt{5} = 6\sqrt{10}$ .

Squaring a surd.

$(\sqrt{a})^{2} = a$ .

$(\sqrt{7})^{2} = 7$ .
$(2\sqrt{3})^{2} = 4 \times 3 = 12$ .

Expanding brackets uses FOIL.

$(\sqrt{3} + 2)(\sqrt{3} - 5)$ :

F: $\sqrt{3} \times \sqrt{3} = 3$ .
O: $\sqrt{3} \times (-5) = -5\sqrt{3}$ .
I: $2 \times \sqrt{3} = 2\sqrt{3}$ .
L: $2 \times (-5) = -10$ .
Combine: $3 - 5\sqrt{3} + 2\sqrt{3} - 10 = -7 - 3\sqrt{3}$ .

Worked qualitative. Compute $(\sqrt{5} + 1)^{2}$ .

FOIL or square-of-a-binomial: $(\sqrt{5})^{2} + 2 \times \sqrt{5} \times 1 + 1^{2}$ .
$= 5 + 2\sqrt{5} + 1 = 6 + 2\sqrt{5}$ .

Edexcel tip. Show every FOIL step. Mark schemes credit method explicitly. After multiplying, simplify any surds and combine like terms.

Same radical: combine.
Different: leave alone.
$\sqrt{a} \times \sqrt{b} = \sqrt{ab}$ .
$(\sqrt{a})^{2} = a$ .
FOIL for brackets.

Rationalising the denominator

Get rid of surds in denominators. Simple: multiply by the surd. Complex: use the conjugate.

Why? Edexcel prefers (and mark schemes expect) answers with rational denominators.

Simple case: denominator is $\sqrt{c}$ .

Multiply numerator AND denominator by $\sqrt{c}$ :

$\dfrac{a}{\sqrt{c}} \times \dfrac{\sqrt{c}}{\sqrt{c}} = \dfrac{a\sqrt{c}}{c}$

Example: $\dfrac{6}{\sqrt{3}} = \dfrac{6\sqrt{3}}{3} = 2\sqrt{3}$ .

Complex case: denominator is $a + b\sqrt{c}$ or $a - b\sqrt{c}$ .

Use the CONJUGATE — same expression with the OTHER sign on the surd.

The product $(a + b\sqrt{c})(a - b\sqrt{c}) = a^{2} - b^{2}c$ has no surd.

Example. $\dfrac{4}{3 + \sqrt{2}}$ .

Multiply by conjugate $\dfrac{3 - \sqrt{2}}{3 - \sqrt{2}}$ :

$\dfrac{4(3 - \sqrt{2})}{(3 + \sqrt{2})(3 - \sqrt{2})} = \dfrac{12 - 4\sqrt{2}}{9 - 2} = \dfrac{12 - 4\sqrt{2}}{7}$

Why does the conjugate work? It's the difference of two squares:

$(a + b\sqrt{c})(a - b\sqrt{c}) = a^{2} - (b\sqrt{c})^{2} = a^{2} - b^{2}c$

The middle terms cancel; what's left is rational.

Worked qualitative. Rationalise $\dfrac{2}{1 - \sqrt{3}}$ .

Conjugate: $1 + \sqrt{3}$ .
$\dfrac{2(1 + \sqrt{3})}{(1 - \sqrt{3})(1 + \sqrt{3})} = \dfrac{2 + 2\sqrt{3}}{1 - 3} = \dfrac{2 + 2\sqrt{3}}{-2} = -1 - \sqrt{3}$ .

Edexcel tip. Show the conjugate explicitly. Multiply both numerator and denominator. Then expand and simplify. Each step earns marks.

Simple: multiply by $\sqrt{c}/\sqrt{c}$ .
Complex: multiply by conjugate.
Conjugate flips the sign on the surd.
Difference of squares kills the surd.

How it’s examined

Surds appear regularly Paper 1H (3-5 marks: simplify or rationalise). Sometimes in algebra/quadratic context. Examiner reports flag (1) writing $\sqrt{a+b} = \sqrt{a} + \sqrt{b}$ , (2) leaving non-simplified surds, (3) wrong sign on conjugate.

Sources: Pearson Edexcel International GCSE Mathematics A 4MA1 specification (2026 onwards); 4MA1 Examiner Reports — Jan and May/Jun 2022-2024 series; Past Papers — 4MA1/1H May/Jun 2024, 4MA1/1H Jan 2024. Last reviewed 2026-05-07.

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Step-by-step worked examples — Surds

Step-by-step solutions to past-paper-style questions on surds, written exactly the way a tutor would explain them at the board.

1Simplify a surd
Higher• simplify, surd
Question
Simplify $\sqrt{72}$ .
Step-by-step solution
1. Step 1
  Find the LARGEST PERFECT SQUARE FACTOR of $72$ . $72 = 36 \times 2$ , and $36$ is a perfect square.
2. Step 2
  Use $\sqrt{ab} = \sqrt{a} \times \sqrt{b}$ :
  $\sqrt{72} = \sqrt{36 \times 2} = \sqrt{36} \times \sqrt{2} = 6\sqrt{2}$
Answer
$6\sqrt{2}$
Examiner tip
ALWAYS pull out the LARGEST perfect square factor. If you do $\sqrt{72} = \sqrt{4 \times 18} = 2\sqrt{18}$ , you'd need to simplify further. Edexcel mark schemes credit the simplest form only.
2Add and subtract surds
Higher• add, subtract
Question
Simplify $\sqrt{50} + \sqrt{18} - \sqrt{8}$ .
Step-by-step solution
1. Step 1
  Simplify each surd: $\sqrt{50} = 5\sqrt{2}$ , $\sqrt{18} = 3\sqrt{2}$ , $\sqrt{8} = 2\sqrt{2}$ .
2. Step 2
  Now all in $\sqrt{2}$ :
  $5\sqrt{2} + 3\sqrt{2} - 2\sqrt{2} = (5 + 3 - 2)\sqrt{2} = 6\sqrt{2}$
Answer
$6\sqrt{2}$
Examiner tip
You can only add/subtract surds with the SAME radical part. Simplify first, then combine 'like terms'.
3Multiply and expand surds
Higher• Adapted from 4MA1/1H May/Jun 2024 Q12• multiply, expand
Question
Expand and simplify $(\sqrt{3} + 2)(\sqrt{3} - 5)$ .
Step-by-step solution
1. Step 1
  FOIL — multiply each pair:
  $\sqrt{3} \times \sqrt{3} - 5\sqrt{3} + 2\sqrt{3} - 10$
2. Step 2
  $\sqrt{3} \times \sqrt{3} = 3$ . Combine like terms:
  $3 - 3\sqrt{3} - 10 = -7 - 3\sqrt{3}$
Answer
$-7 - 3\sqrt{3}$
Examiner tip
FOIL applies to surd brackets just like algebraic ones. $\sqrt{a} \times \sqrt{a} = a$ — the surd disappears.
4Rationalise a simple denominator
Higher• rationalising
Question
Rationalise the denominator of $\dfrac{6}{\sqrt{3}}$ .
Step-by-step solution
1. Step 1
  Multiply numerator AND denominator by $\sqrt{3}$ :
  $\frac{6}{\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}} = \frac{6\sqrt{3}}{3}$
2. Step 2
  Simplify: $\dfrac{6\sqrt{3}}{3} = 2\sqrt{3}$ .
Answer
$2\sqrt{3}$
Examiner tip
$\sqrt{a} \times \sqrt{a} = a$ kills the surd in the denominator. Always simplify the result.
5Rationalise using a conjugate
Higher• Adapted from 4MA1/1H Jan 2024 Q19• rationalising, conjugate
Question
Rationalise the denominator of $\dfrac{4}{3 + \sqrt{2}}$ , giving your answer in the form $a + b\sqrt{2}$ .
Step-by-step solution
1. Step 1
  Multiply by the CONJUGATE: $3 - \sqrt{2}$ .
  $\frac{4}{3 + \sqrt{2}} \times \frac{3 - \sqrt{2}}{3 - \sqrt{2}} = \frac{4(3 - \sqrt{2})}{(3 + \sqrt{2})(3 - \sqrt{2})}$
2. Step 2
  Denominator: difference of two squares $= 3^{2} - (\sqrt{2})^{2} = 9 - 2 = 7$ .
3. Step 3
  Numerator: $4(3 - \sqrt{2}) = 12 - 4\sqrt{2}$ .
4. Step 4
  Combine:
  $\frac{12 - 4\sqrt{2}}{7} = \frac{12}{7} - \frac{4}{7}\sqrt{2}$
Answer
$\dfrac{12}{7} - \dfrac{4}{7}\sqrt{2}$
Examiner tip
The CONJUGATE of $a + b\sqrt{c}$ is $a - b\sqrt{c}$ . Multiplying gives $(a)^{2} - (b\sqrt{c})^{2}$ — a difference of squares with no surd.

Key Formulae — Surds

The formulae you need to memorise for surds on the Pearson Edexcel IGCSE 4MA1 paper, with every variable defined in plain English and a note on when to use it.

Product of surds
$\sqrt{a} \times \sqrt{b} = \sqrt{ab}$
$a, b$
non-negative reals
When to use
Combining or splitting surds.
Example
$\sqrt{8} \times \sqrt{2} = \sqrt{16} = 4$ .
Quotient of surds
$\dfrac{\sqrt{a}}{\sqrt{b}} = \sqrt{\dfrac{a}{b}}$
$a$
non-negative
$b$
positive
When to use
Simplifying ratio of surds.
Example
$\dfrac{\sqrt{50}}{\sqrt{2}} = \sqrt{25} = 5$ .
Square of a surd
$(\sqrt{a})^{2} = a$
$a$
non-negative
When to use
When a surd is multiplied by itself.
Example
$(\sqrt{7})^{2} = 7$ .
Conjugate trick
$(a + b\sqrt{c})(a - b\sqrt{c}) = a^{2} - b^{2}c$
$a, b, c$
any real numbers, $c \ge 0$
When to use
Rationalising denominators of the form $a \pm b\sqrt{c}$ .
Example
$(3 + \sqrt{2})(3 - \sqrt{2}) = 9 - 2 = 7$ .

Key Definitions and Keywords — Surds

Definitions to memorise and the exact keywords mark schemes credit for surds answers — sharpened from recent examiner reports for the 2026 Pearson Edexcel IGCSE 4MA1 sitting.

Surd
Examiner keyword
A square root (or other root) of a non-perfect square (or other non-perfect $n$ th power). Surds are irrational.
Example
$\sqrt{2}, \sqrt{3}, \sqrt{5}$ are surds. $\sqrt{4} = 2$ is NOT a surd (perfect square).
Rationalise (the denominator)
Examiner keyword
Rewrite a fraction so the denominator is a rational number (no surd). Achieved by multiplying numerator and denominator by an appropriate factor.
Example
$\dfrac{1}{\sqrt{3}} = \dfrac{\sqrt{3}}{3}$ .
Conjugate
Examiner keyword
For $a + b\sqrt{c}$ , the conjugate is $a - b\sqrt{c}$ . Multiplying gives a rational result $a^{2} - b^{2}c$ .
Example
Conjugate of $3 + \sqrt{2}$ is $3 - \sqrt{2}$ .
Simplest form (surd)
$a\sqrt{b}$ where $b$ has NO perfect square factors greater than $1$ .
Example
$\sqrt{72} = 6\sqrt{2}$ is in simplest form. $2\sqrt{18}$ is NOT.

Common Mistakes and Misconceptions — Surds

The traps other students keep falling into on surds questions — taken from recent Pearson Edexcel IGCSE 4MA1 examiner reports and mark schemes — and how to avoid them.

✕Writing $\sqrt{a + b} = \sqrt{a} + \sqrt{b}$
Why it happens
Looks like distribution.
How to avoid it
$\sqrt{a + b} \ne \sqrt{a} + \sqrt{b}$ . Test: $\sqrt{9 + 16} = \sqrt{25} = 5$ , but $\sqrt{9} + \sqrt{16} = 3 + 4 = 7$ . NOT the same.
✕Pulling out a small perfect square factor instead of the largest
4MA1/1H — examiner reports 2022-2024
Why it happens
$\sqrt{72} = \sqrt{4 \times 18} = 2\sqrt{18}$ feels right but isn't simplified.
How to avoid it
Find the LARGEST perfect square factor: $72 = 36 \times 2 \to 6\sqrt{2}$ . If you accidentally do $4 \times 18$ , simplify $\sqrt{18}$ further.
✕Adding $\sqrt{3} + \sqrt{5}$ as $\sqrt{8}$
Why it happens
Confusing addition with multiplication.
How to avoid it
Surds with DIFFERENT radicals can't be combined: $\sqrt{3} + \sqrt{5}$ stays as is. Multiplication: $\sqrt{3} \times \sqrt{5} = \sqrt{15}$ .
✕Multiplying ONLY the numerator by $\sqrt{c}$ (not the denominator)
Why it happens
Hurried.
How to avoid it
ALWAYS multiply BOTH parts by the same value. That's just '× 1' in disguise. Mark schemes deduct heavily for one-sided.
✕Using the same sign as the original instead of conjugate
4MA1/1H Jan 2024 — examiner report Q19
Why it happens
Forgetting to flip.
How to avoid it
Conjugate FLIPS the sign on the surd. $a + b\sqrt{c} \to a - b\sqrt{c}$ . Then $(a + b\sqrt{c})(a - b\sqrt{c}) = a^{2} - b^{2}c$ — no surd.

Practice questions

Exam-style questions with step-by-step worked solutions. Try one before checking the method.

Past paper style quiz

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Video lesson

Short walkthrough of the concepts students most often get stuck on.

Surds — frequently asked questions

The things students keep getting wrong in this sub-topic, answered.

Surds

Short Study Notes in the form of Slides

Detailed Study Notes

What you’ll learn

How it’s examined

1Simplify a surd

2Add and subtract surds

3Multiply and expand surds

4Rationalise a simple denominator

5Rationalise using a conjugate

Product of surds

Quotient of surds

Square of a surd

Conjugate trick

Surd

Rationalise (the denominator)

Conjugate

Simplest form (surd)

✕Writing a+b=a+b\sqrt{a + b} = \sqrt{a} + \sqrt{b}a+b​=a​+b​

✕Pulling out a small perfect square factor instead of the largest

✕Adding 3+5\sqrt{3} + \sqrt{5}3​+5​ as 8\sqrt{8}8​

✕Multiplying ONLY the numerator by c\sqrt{c}c​ (not the denominator)

✕Using the same sign as the original instead of conjugate

Practice questions

Past paper style quiz

Assess your understanding

Video lesson

Surds — frequently asked questions

✕Writing $\sqrt{a + b} = \sqrt{a} + \sqrt{b}$

✕Adding $\sqrt{3} + \sqrt{5}$ as $\sqrt{8}$

✕Multiplying ONLY the numerator by $\sqrt{c}$ (not the denominator)