What is N8: Surds and exact values?

An exact answer keeps fractions, multiples of π and, on Higher tier, surds rather than replacing them with rounded decimals. Combine exact fractions with a common denominator and leave circle results as a coefficient of π.

How many marks are N8 questions worth?

In verified past-paper appearances, N8 questions on Edexcel GCSE Maths 1MA1 have been worth 2–6 marks.

Which past exam papers has N8 appeared in?

N8 has appeared in 21 verified past-paper questions indexed on this page, with direct links to the official question paper and mark scheme for each.

Calculate exactly with fractions, surds and multiples of π; simplify surd expressions involving squares (e.g. √12 = √(4 × 3) = √4 × √3 = 2√3) and rationalise denominators

Surds and exact values

Worked answers, methods and verified real exam appearances for N8 on Edexcel GCSE Maths 1MA1.

Explanation

An exact answer keeps fractions, multiples of $\pi$ and, on Higher tier, surds rather than replacing them with rounded decimals. Combine exact fractions with a common denominator and leave circle results as a coefficient of $\pi$ .
Higher-tier surds are simplified by extracting square factors: $\sqrt{12}=\sqrt{4\times3}=2\sqrt3$ .
Like surds can then be collected, and a denominator such as $\sqrt5$ is rationalised by multiplying numerator and denominator by $\sqrt5$ .
Foundation questions do not require surd manipulation, but do require exact fractions and multiples of $\pi$ .
When the command says “exact”, no rounded decimal should appear in the final answer.

Worked example

Higher tier: Simplify $\sqrt{75}-\sqrt{12}$ and give an exact answer.

1.Extract the largest square factor: $\sqrt{75}=\sqrt{25\times3}=5\sqrt3$ .
2.Similarly, $\sqrt{12}=\sqrt{4\times3}=2\sqrt3$ .
3.Collect like surds: $5\sqrt3-2\sqrt3=3\sqrt3$ .

Answer: $3\sqrt3$ .

Common mistakes

Don't replace $\pi$ with $3.14$ when an exact value is required.
Don't write $\sqrt{a+b}=\sqrt a+\sqrt b$ .
Don't collect unlike surds, for example treating $\sqrt2+\sqrt3$ as $2\sqrt5$ .

Exam tip

If the question says “exact”, retain $\pi$ , fractions or surds and simplify them fully rather than using a calculator decimal.

Worked practice

Tier 1 · Easy

Work out

\dfrac{3}{4}+\dfrac{5}{6}

. Give an exact answer.

(2)

(Total for Question 1 is 2 marks)

Mark scheme

Mark scheme for question 1
Question	Answer	Mark	Mark scheme
1	$\dfrac{19}{12}$ $1\dfrac{7}{12}$	2	Use denominator $12$ : $\dfrac{3}{4}=\dfrac{9}{12}$ and $\dfrac{5}{6}=\dfrac{10}{12}$ . Their sum is $\dfrac{19}{12}=1\dfrac{7}{12}$ .

Tier 2 · Standard

A circle has radius

7

cm. Work out its exact area.

(2)

(Total for Question 2 is 2 marks)

Mark scheme

Mark scheme for question 2
Question	Answer	Mark	Mark scheme
2	$49\pi\text{ cm}^2$	2	Use $A=\pi r^2$ . Then $A=\pi\times7^2=49\pi\text{ cm}^2$ .

Tier 3 · Hard

Work out

\dfrac{3\pi}{5}+\dfrac{7\pi}{8}-\dfrac{\pi}{4}

. Give an exact answer.

(3)

(Total for Question 3 is 3 marks)

Mark scheme

Mark scheme for question 3
Question	Answer	Mark	Mark scheme
3	$\dfrac{49\pi}{40}$	3	Use denominator $40$ : $\dfrac{3\pi}{5}=\dfrac{24\pi}{40}$ , $\dfrac{7\pi}{8}=\dfrac{35\pi}{40}$ and $\dfrac{\pi}{4}=\dfrac{10\pi}{40}$ . Therefore the result is $\dfrac{(24+35-10)\pi}{40}=\dfrac{49\pi}{40}$ .

Tier 1 · Easy

Work out

\dfrac{7}{10}-\dfrac{1}{4}

. Give an exact answer.

(2)

(Total for Question 4 is 2 marks)

Mark scheme

Mark scheme for question 4
Question	Answer	Mark	Mark scheme
4	$\dfrac{9}{20}$ (or $0.45$ )	2	Use denominator $20$ : $\dfrac{7}{10}=\dfrac{14}{20}$ and $\dfrac14=\dfrac5{20}$ . The difference is $\dfrac9{20}$ .

Tier 2 · Standard

Simplify

5\pi-\dfrac{7\pi}{4}

(2)

(Total for Question 5 is 2 marks)

Mark scheme

Mark scheme for question 5
Question	Answer	Mark	Mark scheme
5	$\dfrac{13\pi}{4}$	2	Write $5\pi$ in quarters: $5\pi=\dfrac{20\pi}{4}$ . Then $\dfrac{20\pi}{4}-\dfrac{7\pi}{4}=\dfrac{13\pi}{4}$ .

Tier 3 · Hard

Work out

\left(\dfrac{5\pi}{6}+\dfrac{7\pi}{9}\right)\div\dfrac{2}{3}

. Give an exact answer.

(3)

(Total for Question 6 is 3 marks)

Mark scheme

Mark scheme for question 6
Question	Answer	Mark	Mark scheme
6	$\dfrac{29\pi}{12}$	3	First $\dfrac{5\pi}{6}+\dfrac{7\pi}{9}=\dfrac{15\pi}{18}+\dfrac{14\pi}{18}=\dfrac{29\pi}{18}$ . Dividing by $\dfrac23$ means multiplying by $\dfrac32$ , giving $\dfrac{29\pi}{18}\times\dfrac32=\dfrac{29\pi}{12}$ .

Tier 2 · Standard

A ribbon is

5\pi+\dfrac34

metres long. A length of

2\pi-\dfrac16

metres is used. Work out the exact length left.

(3)

(Total for Question 7 is 3 marks)

Mark scheme

Mark scheme for question 7
Question	Answer	Mark	Mark scheme
7	$3\pi+\dfrac{11}{12}$ metres	3	Subtract the used length: $5\pi+\dfrac34-(2\pi-\dfrac16)=3\pi+\dfrac34+\dfrac16$ . Using denominator $12$ gives $3\pi+\dfrac9{12}+\dfrac2{12}=3\pi+\dfrac{11}{12}$ metres.

Tier 3 · Hard

A wheel has radius

0.35

metres. It makes

24

complete rotations. Work out the exact distance travelled.

(3)

(Total for Question 8 is 3 marks)

Mark scheme

Mark scheme for question 8
Question	Answer	Mark	Mark scheme
8	$\dfrac{84\pi}{5}$ metres (or $16.8\pi$ metres)	3	One rotation covers the circumference $2\pi\times0.35=0.7\pi$ metres. In $24$ rotations the distance is $24\times0.7\pi=16.8\pi=\dfrac{84\pi}{5}$ metres.

Tier 3 · Hard

Higher only: Work out

\left(\sqrt{50}-\sqrt{8}\right)\left(\sqrt{18}+\sqrt{2}\right)

. Give an exact answer.

(4)

(Total for Question 9 is 4 marks)

Mark scheme

Mark scheme for question 9
Question	Answer	Mark	Mark scheme
9	$24$	4	$\sqrt{50}=5\sqrt2$ , $\sqrt8=2\sqrt2$ and $\sqrt{18}=3\sqrt2$ . The expression becomes $(3\sqrt2)(4\sqrt2)=12\times2=24$ .

Q10

Tier 3 · Hard

Higher only: Rationalise the denominator and simplify

\dfrac{4}{\sqrt{7}-1}-\sqrt{7}

(4)

(Total for Question 10 is 4 marks)

Mark scheme

Mark scheme for question 10
Question	Answer	Mark	Mark scheme
10	$\dfrac{2-\sqrt{7}}{3}$	4	Multiply the fraction by $\dfrac{\sqrt7+1}{\sqrt7+1}$ to get $\dfrac{4(\sqrt7+1)}{7-1}=\dfrac{2\sqrt7+2}{3}$ . Subtracting $\sqrt7=\dfrac{3\sqrt7}{3}$ gives $\dfrac{2-\sqrt7}{3}$ .

Verified exam appearances

Series	Paper	Question	Marks	Calculator	Tier	Links
2024-06	2H	Q15	2	Allowed	Higher	QP MS
2019-06	1H	Q14	2	Non-calculator	Higher	QP MS
2022-11	1H	Q23	5	Non-calculator	Higher	QP MS
2022-06	1H	Q21	5	Non-calculator	Higher	QP MS
2023-06	1H	Q23	4	Non-calculator	Higher	QP MS
2023-11	1H	Q20	4	Non-calculator	Higher	QP MS
2021-11	1F	Q26	4	Non-calculator	Foundation	QP MS
2023-06	1H	Q16	4	Non-calculator	Higher	QP MS
2024-06	1H	Q17	3	Non-calculator	Higher	QP MS
2019-06	1F	Q19	4	Non-calculator	Foundation	QP MS
2024-11	1H	Q14	3	Non-calculator	Higher	QP MS
2022-06	1F	Q22	3	Non-calculator	Foundation	QP MS
2024-11	1H	Q16	6	Non-calculator	Higher	QP MS
2021-11	1H	Q19	4	Non-calculator	Higher	QP MS
2019-11	1H	Q16	5	Non-calculator	Higher	QP MS
2022-11	1H	Q15	3	Non-calculator	Higher	QP MS
2024-06	1H	Q15	4	Non-calculator	Higher	QP MS
2019-06	1H	Q18	5	Non-calculator	Higher	QP MS
2023-11	1H	Q22	4	Non-calculator	Higher	QP MS
2019-06	1H	Q15	4	Non-calculator	Higher	QP MS
2021-11	1H	Q7	4	Non-calculator	Higher	QP MS

Other points in N Number

Want help turning this into marks?

Bring N8 or any tricky specification point, and we can work through the method and exam wording together.

Book a free intro call