Know square and cube numbers, find square and cube roots, and understand what the notation means. It is a Foundation topic, aimed at grade 1. Free questions with answers are on this page. Worked examples show each line before you try the questions.
A power is repeated multiplication: 112=11×11=121 and 25=2×2×2×2×2=32. The square root undoes a square: 121=11 (the positive value, unless a ± is asked). The cube root undoes a cube: 364=4 because 43=64.
Roots and powers sit on the same BIDMAS rung as each other, above multiply and divide. 9+16 is 25=5, not 3+4, because the addition is inside the root. 9+16 would be 3+4=7 — different question.
Learn the square numbers to 152=225 and the cubes of 1 to 5 and 10. They turn up constantly on the non-calculator paper.
Objectives
Read square and cube notation.
Work out a small power.
Find a whole-number square root.
Key points
Recall square numbers up to 152=225 and cube numbers 1,8,27,64,125,1000.
Work out powers such as 23 and 34 by repeated multiplication.
Find square roots and cube roots of perfect squares and cubes without a calculator.
Estimate a square root that is not exact by finding the two square numbers it lies between.
Worked examples
Example 1
Work out 7².
2 marks
The power is 2, so there are two factors of 7.
7 × 7 = 49.
7² = 49.
Answer: 49
Common slips
Slip: Multiply 7 by 2. The 2 counts the factors. It is not the other factor.
Slip: 7 × 2 = 14. Fourteen is double 7, not the square.
Slip: 7² = 14. A square of a number above 2 is larger than its double.
Example 2
Work out 2³ × 5.
2 marks
2³ = 2 × 2 × 2 = 8.
8 × 5 = 40.
2³ × 5 = 40.
Answer: 40
Common slips
Slip: 2³ = 6. The power 3 means three factors, not 2 × 3.
Slip: 8 × 5 = 13. This is multiplication, not addition.
Slip: 2³ × 5 = 30. 3 × 2 × 5 is a different expression.
Example 3
A square tile has area 81 cm². What is the side length in centimetres?
3 marks
The area is the side squared.
√81 = 9, because 9 × 9 = 81.
The side is 9 cm.
Answer: 9
Common slips
Slip: Halve 81. A square root is not a half.
Slip: √81 = 9 × 9. 9 × 9 is the area again, not the side.
Slip: The side is 40.5 cm. 40.5 is half of 81.
Questions and answers
GCSE Maths powers & roots questions and answers. Try each one, then open the answer. Easy questions come first.
Question 1 Easy2 marks
Work out 6².
Show answer
Answer: 36
6 × 6 = 36.
The power is 2.
M1 — 6 × 6 = 36.
A1 — Answer 36
Question 2 Easy2 marks
Work out 10².
Show answer
Answer: 100
10 × 10 = 100.
The power is 2.
M1 — 10 × 10 = 100.
A1 — Answer 100
Question 3 Medium2 marks
Work out 9².
Show answer
Answer: 81
9 × 9 = 81.
The power is 2.
M1 — 9 × 9 = 81.
A1 — Answer 81
Question 4 Medium2 marks
What is √64?
Show answer
Answer: 8
8 × 8 = 64.
The square root is 8.
M1 — 8 × 8 = 64.
A1 — Answer 8
Question 5 Hard2 marks
A square garden has area 64 m². What is the side length?
Show answer
Answer: 8
√64 = 8.
The side is 8 m.
M1 — √64 = 8.
A1 — Answer 8
Question 6 Hard2 marks
Work out 3³ + 4².
Show answer
Answer: 43
3³ = 27 and 4² = 16.
27 + 16 = 43.
M1 — 3³ = 27 and 4² = 16.
A1 — Answer 43
Question 7 Exam-style3 marks
A cube has side 5 cm. How many 1 cm cubes fill it? That volume is 5³.
Show answer
Answer: 125
5 × 5 = 25.
25 × 5 = 125.
M1 — Squares the side
M1 — Multiplies by the side again
A1 — Answer 125
Common mistakes
Most lost marks on this topic are one slip.
Multiplying by the power, so 6² becomes 12.
The power counts how many times the base is a factor. 6² = 6 × 6.
Halving to find a square root.
Ask which number times itself gives the square.
Doing 2³ as 2 × 2 and stopping.
A cube needs three factors: 2 × 2 × 2.
Spot the slip
What does 4² mean?
4 × 4
4 × 2
4 + 4
Show the slip
4 × 4 The small 2 means use 4 as a factor twice.
Exam tips
Be ready to do this without a calculator. Paper 1 is the non-calculator paper.
If the question says “show that”, the result is given. The marks are for the steps that reach it.
The method is the same on AQA-style and Edexcel-style papers.
Learn the squares to 15 and the cubes 1, 8, 27, 64, 125, 1000 by heart — they appear in many non-calculator questions.
Show the repeated multiplication (3×3×3=27) so the method mark is safe even if you slip.
Printable worksheet
The same questions are in a PDF you can print. Answers start on a later page.