What is N6: Powers and roots?

A positive integer power represents repeated multiplication: a^n contains n factors equal to a.

How many marks are N6 questions worth?

In verified past-paper appearances, N6 questions on Edexcel GCSE Maths 1MA1 have been worth 1–5 marks.

Which past exam papers has N6 appeared in?

N6 has appeared in 20 verified past-paper questions indexed on this page, with direct links to the official question paper and mark scheme for each.

Use positive integer powers and associated real roots (square, cube and higher), recognise powers of 2, 3, 4, 5; estimate powers and roots of any given positive number

Powers and roots

Worked answers, methods and verified real exam appearances for N6 on Edexcel GCSE Maths 1MA1.

Explanation

A positive integer power represents repeated multiplication: $a^n$ contains $n$ factors equal to $a$ .
An associated root reverses that power, so $\sqrt{81}=9$ , $\sqrt[3]{125}=5$ , and $\sqrt[4]{16}=2$ .
Learn common powers of $2$ , $3$ , $4$ and $5$ because they make exact roots quickly recognisable.
Higher tier: to estimate a root that is not exact, place its radicand between nearby known powers; for example, $3^3<40<4^3$ shows $3<\sqrt[3]{40}<4$ .
In an exam, distinguish an exact evaluation from an estimate and clearly state the bounding powers used.

Worked example

Higher tier: Estimate $\sqrt[3]{200}$ to the nearest whole number without a calculator.

1.Use nearby cubes: $5^3=125$ and $6^3=216$ , so $5<\sqrt[3]{200}<6$ .
2.The rounding threshold is $5.5^3=166.375$ ; since $200>166.375$ , the cube root is above $5.5$ .
3.Round the estimate to the nearest whole number.

Answer: $\sqrt[3]{200}\approx6$ .

Common mistakes

Don't calculate $3^4$ as $3\times4$ instead of four factors of $3$ .
Don't use a square root when the inverse operation required is a cube root.
Don't state an estimated root as an exact equality.

Exam tip

For an estimate, write the two consecutive known powers that bound the radicand before choosing the nearer root.

Worked practice

Tier 1 · Easy

Work out

2^5

(1)

(Total for Question 1 is 1 mark)

Mark scheme

Mark scheme for question 1
Question	Answer	Mark	Mark scheme
1	$32$	1	$2^5=2\times2\times2\times2\times2=32$ .

Tier 2 · Standard

Work out

\sqrt[3]{343}

(1)

(Total for Question 2 is 1 mark)

Mark scheme

Mark scheme for question 2
Question	Answer	Mark	Mark scheme
2	$7$	1	Since $7^3=7\times7\times7=343$ , the associated cube root is $\sqrt[3]{343}=7$ .

Tier 3 · Hard

Work out

\sqrt[4]{1296}+\sqrt[3]{512}

(3)

(Total for Question 3 is 3 marks)

Mark scheme

Mark scheme for question 3
Question	Answer	Mark	Mark scheme
3	$14$	3	Because $6^4=1296$ , $\sqrt[4]{1296}=6$ . Because $8^3=512$ , $\sqrt[3]{512}=8$ . Their sum is $6+8=14$ .

Tier 1 · Easy

Write down the two consecutive whole numbers between which

\sqrt{30}

lies.

(1)

(Total for Question 4 is 1 mark)

Mark scheme

Mark scheme for question 4
Question	Answer	Mark	Mark scheme
4	$5$ and $6$	1	$5^2=25$ and $6^2=36$ . Since $25<30<36$ , $\sqrt{30}$ lies between $5$ and $6$ .

Tier 2 · Standard

Write

625

as a power of

5

(1)

(Total for Question 5 is 1 mark)

Mark scheme

Mark scheme for question 5
Question	Answer	Mark	Mark scheme
5	$5^4$	1	$625=5\times125=5\times5\times25=5\times5\times5\times5$ , so $625=5^4$ .

Tier 3 · Hard

Work out

\sqrt[3]{1000}\times\sqrt{196}\div\sqrt[3]{8}

(3)

(Total for Question 6 is 3 marks)

Mark scheme

Mark scheme for question 6
Question	Answer	Mark	Mark scheme
6	$70$	3	$\sqrt[3]{1000}=10$ , $\sqrt{196}=14$ and $\sqrt[3]{8}=2$ . Therefore $10\times14\div2=70$ .

Tier 2 · Standard

Write

4096

as a power of

4

. Hence work out

\sqrt[6]{4096}

(3)

(Total for Question 7 is 3 marks)

Mark scheme

Mark scheme for question 7
Question	Answer	Mark	Mark scheme
7	$4096=4^6$ and $\sqrt[6]{4096}=4$	3	$4^2=16$ , $4^3=64$ and $4^6=64^2=4096$ . Therefore the number whose sixth power is $4096$ is $4$ , so $\sqrt[6]{4096}=4$ .

Tier 3 · Hard

a

and

b

are positive integers.

a^4=625

and

\sqrt[3]{b}=a+3

. Work out the value of

b

(3)

(Total for Question 8 is 3 marks)

Mark scheme

Mark scheme for question 8
Question	Answer	Mark	Mark scheme
8	$b=512$	3	Since $5^4=625$ and $a$ is positive, $a=5$ . Then $\sqrt[3]{b}=5+3=8$ , so $b=8^3=512$ .

Tier 3 · Hard

Higher only: The positive number

x

satisfies

x^4=580

. Given that

4.9^4=576.4801

, use a suitable comparison to work out

x

to the nearest tenth.

(4)

(Total for Question 9 is 4 marks)

Mark scheme

Mark scheme for question 9
Question	Answer	Mark	Mark scheme
9	$x=4.9$	4	$4.9^4=576.4801<580$ , so $x>4.9$ . At the rounding boundary, $4.95^2=24.5025>24.5$ , so $4.95^4>24.5^2=600.25>580$ . Hence $x<4.95$ , and $x$ rounds to $4.9$ to $1$ decimal place.

Q10

Tier 3 · Hard

n

is a positive integer that is both a square number and a cube number. It satisfies

\sqrt[3]{n}+\sqrt{n}=150

. Work out the value of

n

(4)

(Total for Question 10 is 4 marks)

Mark scheme

Mark scheme for question 10
Question	Answer	Mark	Mark scheme
10	$n=15\,625$	4	Because $n$ is both a square and a cube, write $n=a^6$ for a positive integer $a$ . Then $\sqrt[3]{n}=a^2$ and $\sqrt n=a^3$ , so $a^2+a^3=150$ . Since $5^2+5^3=25+125=150$ , $a=5$ and $n=5^6=15\,625$ .

Verified exam appearances

Series	Paper	Question	Marks	Calculator	Tier	Links
2021-11	2F	Q6	2	Allowed	Foundation	QP MS
2019-11	2F	Q4	1	Allowed	Foundation	QP MS
2019-06	3F	Q4	1	Allowed	Foundation	QP MS
2019-11	2F	Q5	1	Allowed	Foundation	QP MS
2023-06	1F	Q22	4	Non-calculator	Foundation	QP MS
2023-06	1H	Q3	4	Non-calculator	Higher	QP MS
2022-11	1F	Q2	1	Non-calculator	Foundation	QP MS
2023-11	2F	Q4	1	Allowed	Foundation	QP MS
2024-11	3F	Q5	1	Allowed	Foundation	QP MS
2022-06	3H	Q9	4	Allowed	Higher	QP MS
2024-06	1F	Q11	3	Non-calculator	Foundation	QP MS
2024-06	1F	Q5	1	Non-calculator	Foundation	QP MS
2021-11	1H	Q9	4	Non-calculator	Higher	QP MS
2023-11	1H	Q2	4	Non-calculator	Higher	QP MS
2019-06	1F	Q15	2	Non-calculator	Foundation	QP MS
2019-06	1H	Q8	4	Non-calculator	Higher	QP MS
2019-11	2F	Q29	5	Allowed	Foundation	QP MS
2021-11	1F	Q5	1	Non-calculator	Foundation	QP MS
2019-11	2H	Q9	5	Allowed	Higher	QP MS
2024-11	2F	Q8	2	Allowed	Foundation	QP MS

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