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Powers and roots

Free Printable Powers and roots PDF Worksheets, Revision Guide & GCSE Past Paper Questions (Grade 1)

Download free powers and roots worksheets (PDF), GCSE maths past paper questions and worked solutions for Powers and roots (1K). This is a Grade 1 number topic from the GCSE questions by topic collection for Foundation and Higher tier GCSE Maths (Edexcel, AQA, OCR and IGCSE).

The free PDF worksheets table below brings together printable question booklets (Qs) and worked answers or mark schemes (Ans) from Maths Genie, Corbett Maths, Maths Takeaway, Dr Austin, Maths4EveryOne, GCSE Maths Rocket, E Classroom and PMT, along with past paper question booklets.

Below the worksheet links you can read our powers and roots revision notes and worked examples with diagrams, try real GCSE past paper powers and roots questions with step-by-step solutions, or download a PDF of all powers and roots worksheet links.

Free Powers and roots PDF Worksheets & Answers

Grade 1 • Topic 1K
Resource Questions Answers
GCSE past paper questions & solutions
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Graded exam questions & solutions
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Practice Questions & answers
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Textbook Exercise & answers
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GCSE Maths Takeaway exam questions
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Practice grid / worksheet & answers
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GCSE 9–1 practice booklet & solutions
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Practice questions & answers
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GCSE topic booklet & answers
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PMT (Edxl)
Edexcel GCSE topic questions & solutions
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PMT (AQA)
AQA GCSE topic questions & solutions
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PMT (IGCSE)
Edexcel IGCSE topic questions & solutions
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Interactive 10-question revision grid
 Grid
Online topic notes & examples
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Interactive online lesson & practice
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Self-marking online worksheet
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Grade 1 topics
1A - Addition1B - Subtraction
1C - Multiplication1D - Division
1E - Time1F - Ordering FDP
1G - Place Value1H - Rounding
1J - Negative numbers1K - Powers and roots
1L - BIDMAS1M - Factors, multiples and primes
1N - Coordinates1P - Pictograms
1Q - 2D shapes1R - 3D shapes
GCSE Maths Grade 1 Revision Notes & Method

Guide to Powers and roots: Explanation, Diagrams & Worked Examples

Download Printable One Page Guide (PDF)

Explanation:

A square number (n2) is the result of multiplying an integer by itself (first ten: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100). A cube number (n3) comes from multiplying an integer by itself three times: n × n × n (first five: 1, 8, 27, 64, 125). Never multiply the base by the power (43 ≠ 12).

Roots are the inverse of powers: the square root (√x) asks what positive number squared gives x, and the cube root (∛x) asks what number cubed gives x. Also remember that any non-zero number to the power of 0 equals 1 (x0 = 1).

Example 1: Work out the value of 72 + √81 − 23

Solution:

Evaluate each power or root separately first:
• 72 = 7 × 7 = 49
• √81 = 9  (since 9 × 9 = 81)
• 23 = 2 × 2 × 2 = 8
Combine: 49 + 9 − 8 = Answer: 50

Example 2: Work out the value of 43 − ∛64 + 103

Solution:

Evaluate each term carefully:
• 43 = 4 × 4 × 4 = 64
• ∛64 = 4  (since 4 × 4 × 4 = 64)
• 103 = 10 × 10 × 10 = 1000
Combine: 64 − 4 + 1000 = Answer: 1060
Scanned GCSE Exam Questions & Worked Solutions

GCSE Past Paper Powers and roots Questions & Answers

Past Paper Example 1 AQA Nov 2023 3F Q22 / 3H Q7 3 marks
Full Paper (PDF) • Mark Scheme (PDF)
The first two cube numbers are 1 and 8
=++
Show that
the 3rd cube number can be written as the sum of three different prime numbers.
Worked Solution & Mark Scheme (3 marks)
3rd cube number = 3³ = 27. M2
27 = 3 + 5 + 19 (or 3 + 7 + 17 or 3 + 11 + 13 or 5 + 11 + 11 — must be different primes: 3 + 5 + 19, 3 + 7 + 17, or 3 + 11 + 13). C1
273519=++
Past Paper Example 2 GCSE Nov 2021 1H Q10 4 marks
Full Paper (PDF) • Mark Scheme (PDF)
The diagram shows a shape made from 6 identical squares.
The total area of the shape is 5406 cm²
(a) Find an estimate for the length of one side of each square.
Give your answer correct to the nearest whole number. [3 marks]
(b) Is your answer to part (a) an underestimate or an overestimate?
You must give a reason for your answer. [1 mark]
Worked Solution & Mark Scheme (4 marks)
Part (a) Solution:
5406 ÷ 6 = 901 ≈ 900 M2
√900 = 30 cm A1
Answer (a): 30 cm
Part (b) Solution:
√900 < √901 so 30 is an underestimate C1
Answer (b): Underestimate because 5406 was rounded down to 5400 (or 901 was rounded down to 900)
Past Paper Example 3 GCSE Jun 2025 3F Q10 1 mark
Full Paper (PDF) • Mark Scheme (PDF)
Jason says,
"The cube root of 27 is 9 because 27 divided by 3 is 9"
Is Jason correct?
You must give a reason for your answer.
Worked Solution & Mark Scheme (1 mark)
No, because 3 × 3 × 3 = 27 so the cube root of 27 is 3. Finding a cube root is not dividing by 3. B1
Answer: No with reason (cube root of 27 is 3, not 9)

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