GCSE Maths · Number

Surds GCSE Questions, Worked Examples and Answers

Q: What is a surd?

A surd is an irrational root written exactly, such as √2. A root that simplifies to a rational number, such as √9 = 3, is not a surd.

Q: Can I add different surds?

Only like surds can be collected. For example, 2√3 + 5√3 = 7√3, but √2 + √3 stays as written.

Q: What does rationalise the denominator mean?

Rewrite an equivalent fraction with no surd in its denominator.

Q: Why use a conjugate?

Multiplying (a + √b)(a − √b) gives a² − b, so the surd middle terms cancel.

A surd is a root that cannot be written exactly as a whole number or fraction, such as √2. Keeping the root symbol preserves the exact value; GCSE questions then ask you to simplify and calculate with that exact form.

Edexcel · AQA · OCRHigher tier15 original questions

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Square roots before surd rules

What you need to know about GCSE surds

The square-root symbol √ asks: which positive number multiplied by itself gives the number inside the root? Because 3 × 3 = 9, √9 = 3. No whole number or fraction squares to exactly 2, so √2 is left in root form. A root that cannot be written exactly as a whole number or fraction is called a surd.

Square factor × remaining root

Why √2 is exact but 1.414… is only an approximation

Compare 2 with nearby square numbers. This locates √2 first; a calculator decimal can then approximate it without replacing the exact value.

Nearby squares1² < 2 < 2²2 lies between 1 and 4

Locate the root1 < √2 < 2√2 is not a whole number

Keep it exact√2 ≈ 1.414√2 is exact; the decimal is rounded

From problem to method

Typical problems you need to be able to solve

These are the main kinds of question in this topic. Each card first states the problem, then explains how to solve it.

Simplify a surd

What the problem asks: Write a root such as √72 in the form a√b, where b has no square factor greater than 1.

How to solve it: Split the number into its largest square factor and the remaining factor. The square factor can then come out of the root.

$\sqrt{72}=\sqrt{36\times2}=6\sqrt2$

Add or subtract surds

What the problem asks: Terms may become the same root after simplifying.

How to solve it: Simplify first, then collect like surds exactly as 2x + 5x = 7x: for example, 2√3 + 5√3 = 7√3.

Multiply or expand

What the problem asks: Surds are multiplied or appear in brackets.

How to solve it: Multiply coefficients and roots, expand every bracket term, then simplify roots and collect like terms.

Rationalise one surd

What the problem asks: The denominator contains a single square root.

How to solve it: Multiply top and bottom by that root. This multiplies the fraction by 1, while √3 × √3 = 3 makes the denominator rational.

Rationalise a binomial

What the problem asks: The denominator has the form a + √b or a − √b.

How to solve it: Multiply by the conjugate. The identity (a + b)(a − b) = a² − b² squares the surd and cancels the middle terms.

A reliable routine

Method for simplifying square-root surds

Use this routine when a square root is not already in simplest form. It is not a method for every surds problem; addition and rationalising need the separate choices above.

List or recognise square factors of the number under the root.
Choose the largest square factor to avoid repeated simplification.
Split the root into the square factor and the remaining factor.
Evaluate the square root of the square factor and check that no square factor remains inside the root.

Check: Do not split addition inside a root: √(a + b) is generally not √a + √b. On a non-calculator paper, an exact answer such as 6√2 is required; a rounded decimal such as 8.485… is not equivalent to exact form.

Fully worked

Surds GCSE worked examples

Each solution explains the clue, why the method fits and how to check the result.

Example 1

Simplify a surd

2 marks

Question

Simplify $\sqrt{72}$ .

The largest square factor of $72$ is $36$ .

$\sqrt{72}=\sqrt{36\times2}$

$=\sqrt{36}\sqrt2$

$\boxed{6\sqrt2}$

Example 2

Collect like surds

3 marks

Question

Simplify $3\sqrt{12}+\sqrt{27}$ .

Simplify each root first.

$3\sqrt{12}=3\sqrt{4\times3}=6\sqrt3$

$\sqrt{27}=\sqrt{9\times3}=3\sqrt3$

Now the roots match.

$6\sqrt3+3\sqrt3=\boxed{9\sqrt3}$

Example 3

Use conjugate brackets

3 marks

Question

Expand and simplify $(\sqrt5+2)(\sqrt5-2)$ .

The middle terms cancel because the brackets are conjugates.

$(\sqrt5+2)(\sqrt5-2)=(\sqrt5)^2-2^2$

$=5-4$

$\boxed{1}$

Example 4

Rationalise a single-surd denominator

3 marks

Question

Rationalise the denominator of $\dfrac7{\sqrt3}$ .

Multiply top and bottom by $\sqrt3$ .

$\frac7{\sqrt3}\times\frac{\sqrt3}{\sqrt3}=\frac{7\sqrt3}{3}$

$\boxed{\frac{7\sqrt3}{3}}$

Example 5

Rationalise a binomial denominator

4 marks

Question

Write $\dfrac5{2+\sqrt3}$ with a rational denominator.

Use the conjugate $2-\sqrt3$ .

$\frac5{2+\sqrt3}\times\frac{2-\sqrt3}{2-\sqrt3}$

$=\frac{5(2-\sqrt3)}{2^2-(\sqrt3)^2}$

$=\frac{10-5\sqrt3}{4-3}$

$\boxed{10-5\sqrt3}$

Example 6

Use surds in geometry

4 marks

Question

A right-angled triangle has shorter sides $2$ cm and $3$ cm. Find the exact length of the hypotenuse.

Use Pythagoras’ theorem.

$c^2=2^2+3^2$

$c^2=4+9$

$c^2=13$

Length is positive, so

$\boxed{c=\sqrt{13}\text{ cm}}$

Example 7

Simplify a root of a fraction

3 marks

Question

Simplify $\sqrt{\dfrac{8}{25}}$ .

Split the numerator and denominator.

$\sqrt{\frac{8}{25}}=\frac{\sqrt8}{\sqrt{25}}$

$=\frac{\sqrt{4\times2}}{5}$

$\boxed{\frac{2\sqrt2}{5}}$

Example 8

Use surds after completing the square

5 marks

Question

Solve $x^2-2x-6=0$ exactly by completing the square.

$x^2-2x-6=(x-1)^2-1-6$

$(x-1)^2-7=0$

$(x-1)^2=7$

$x-1=\pm\sqrt7$

$\boxed{x=1\pm\sqrt7}$

15 original questions · total 44 marks

Surds GCSE exam-style questions

Try each question before opening its fully worked answer. The difficulty rises through the set.

Before you startAllow about 50 minutes · leave exact answers in their simplest surd form · answers start collapsed

Identify a surd

1 mark

Which of $\sqrt9$ , $\sqrt{10}$ and $\sqrt{0.25}$ is a surd?

Show worked answer

$\sqrt9=3$ and $\sqrt{0.25}=0.5$ are rational.

$\boxed{\sqrt{10}\text{ is the surd}}$

Simplify a square root

2 marks

Simplify $\sqrt{98}$ .

Show worked answer

$\sqrt{98}=\sqrt{49\times2}$

$=\boxed{7\sqrt2}$

Subtract like surds

3 marks

Simplify $\sqrt{45}-\sqrt5$ .

Show worked answer

$\sqrt{45}=\sqrt{9\times5}=3\sqrt5$

$3\sqrt5-\sqrt5=\boxed{2\sqrt5}$

Multiply and simplify

3 marks

Simplify $2\sqrt3\times5\sqrt6$ .

Show worked answer

$2\sqrt3\times5\sqrt6=10\sqrt{18}$

$=10\times3\sqrt2$

$\boxed{30\sqrt2}$

Simplify a fraction containing several surds

3 marks

Simplify $\dfrac{\sqrt8+\sqrt{18}}{\sqrt2}$ .

Show worked answer

Simplify the roots in the numerator.

$\sqrt8=2\sqrt2$

$\sqrt{18}=3\sqrt2$

$\frac{\sqrt8+\sqrt{18}}{\sqrt2}=\frac{5\sqrt2}{\sqrt2}$

$=\boxed{5}$

Square a surd binomial

2 marks

Expand and simplify $(3+\sqrt2)^2$ .

Show worked answer

$(3+\sqrt2)^2=9+3\sqrt2+3\sqrt2+2$

$=\boxed{11+6\sqrt2}$

Use difference of two squares

3 marks

Simplify $(\sqrt7+\sqrt5)(\sqrt7-\sqrt5)$ .

Show worked answer

$(\sqrt7)^2-(\sqrt5)^2$

$=7-5$

$=\boxed{2}$

Rationalise one root

3 marks

Rationalise the denominator of $\dfrac4{\sqrt5}$ .

Show worked answer

$\frac4{\sqrt5}\times\frac{\sqrt5}{\sqrt5}$

$=\boxed{\frac{4\sqrt5}{5}}$

Rationalise a fraction with surds on top and bottom

4 marks

Write $\dfrac{\sqrt5+1}{\sqrt5-1}$ with a rational denominator.

Show worked answer

Multiply by the conjugate of the denominator, $\sqrt5+1$ .

$\frac{\sqrt5+1}{\sqrt5-1}\times\frac{\sqrt5+1}{\sqrt5+1}$

$=\frac{(\sqrt5+1)^2}{(\sqrt5)^2-1^2}$

$=\frac{6+2\sqrt5}{4}$

$\boxed{\frac{3+\sqrt5}{2}}$

Solve an equation containing surds

3 marks

Find $k$ if $k\sqrt{12}=18\sqrt3$ .

Show worked answer

Simplify $\sqrt{12}$ first.

$k(2\sqrt3)=18\sqrt3$

Divide both sides by $\sqrt3$ .

$2k=18$

$\boxed{k=9}$

Show an exact identity

3 marks

Show that $(2+\sqrt3)^2=7+4\sqrt3$ .

Show worked answer

Expand every term.

$(2+\sqrt3)^2=4+2\sqrt3+2\sqrt3+3$

$=7+4\sqrt3$

as required.

Simplify a surd fraction under one root

3 marks

Simplify $\sqrt{\dfrac{18}{49}}$ .

Show worked answer

$\sqrt{\frac{18}{49}}=\frac{\sqrt{18}}7$

$=\frac{\sqrt{9\times2}}7$

$\boxed{\frac{3\sqrt2}{7}}$

Expand brackets with different surds

4 marks

Expand and simplify $(2+\sqrt3)(4-\sqrt5)$ .

Show worked answer

Multiply every term in the first bracket by every term in the second.

$(2+\sqrt3)(4-\sqrt5)=8-2\sqrt5+4\sqrt3-\sqrt{15}$

The surds are unlike, so they cannot be collected.

$\boxed{8+4\sqrt3-2\sqrt5-\sqrt{15}}$

Find an exact diagonal

3 marks

A square has side length $5$ cm. Find the exact length of its diagonal.

Show worked answer

Use Pythagoras’ theorem.

$d^2=5^2+5^2$

$d^2=50$

$d=\sqrt{50}$

$\boxed{d=5\sqrt2\text{ cm}}$

Rationalise a conjugate quotient

4 marks

Write $\dfrac{2+\sqrt3}{2-\sqrt3}$ with a rational denominator.

Show worked answer

Multiply top and bottom by $2+\sqrt3$ .

$\frac{2+\sqrt3}{2-\sqrt3}\times\frac{2+\sqrt3}{2+\sqrt3}$

$=\frac{(2+\sqrt3)^2}{4-3}$

$=\frac{7+4\sqrt3}{1}$

$\boxed{7+4\sqrt3}$

Examiner-style feedback

Common surds mistakes

Splitting a sum inside a root

The product rule can split √(ab), but √(a + b) does not equal √a + √b in general.

Adding unlike roots

2√3 + 5√2 cannot be combined. Simplify each root first and collect only matching surds.

Using a non-square factor

Taking a factor outside a square root is valid only when its square is a factor of the number inside.

Changing only the denominator

Rationalising multiplies the fraction by a form of 1, so the numerator and denominator must both be multiplied.

30-second recap

Choose, carry out, check

Translate the wording before you reach for a rule.

Keep irrational roots exact.
Take out the largest square factor.
Simplify before collecting like surds.
Use the root or conjugate to rationalise a denominator.

Quick answers

Surds FAQ

What is a surd?

A surd is an irrational root written exactly, such as √2. A root that simplifies to a rational number, such as √9 = 3, is not a surd.

Can I add different surds?

Only like surds can be collected. For example, 2√3 + 5√3 = 7√3, but √2 + √3 stays as written.

What does rationalise the denominator mean?

Rewrite an equivalent fraction with no surd in its denominator.

Why use a conjugate?

Multiplying (a + √b)(a − √b) gives a² − b, so the surd middle terms cancel.

Build connected skills

What to revise next

Exact roots

Completing the square

Solve quadratics whose exact roots contain surds.

Revise completing the square

Power rules

Indices

Connect square roots to fractional indices.

Revise indices

Exact lengths

Trigonometry

Keep geometric answers exact when roots appear.

Revise trigonometry

Content standards

Curriculum and rights review

Curriculum references checked 4 September 2026. Calculating exactly with surds, simplifying roots and rationalising denominators are shared Higher-tier GCSE Mathematics content. All questions, values and solution wording are original Pass an Exam material.

Official specification references