Practise 28 exam-style questions on Powers and roots for Edexcel GCSE Maths (Non-calculator). Write your answer as you would in the exam and get it marked in seconds against the mark scheme, point by point, with an AI tutor on hand if you get stuck.
28 questions
1–5 marks each
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20 questions, marked as you go
Sample questions with model answers
Here are 4 of the questions in this topic, each with a model answer, an explanation of why it earns the marks and the mark scheme an examiner would use.
Question 1
2 marks
Estimate the value of \(\sqrt{50}\). Give your answer correct to 1 decimal place.
Model answer
\(7^2 = 49\) and \(8^2 = 64\) 50 is just above 49, so \(\sqrt{50}\) is just above 7. \(\sqrt{50} \approx 7.1\)
Explanation
Find the square numbers either side of 50. Since 50 is much closer to 49 than to 64, the square root is only slightly more than 7. Check: \(7.1^2 = 50.41\), which is very close to 50.
Mark scheme
1 mark
M1 for \(7^2 = 49\) and \(8^2 = 64\), or a statement that \(\sqrt{50}\) is between 7 and 8 and closer to 7
correct answer with no working scores full marks
1 mark
A1 for an answer in the range 7.0 to 7.1
Question 2
2 marks
Sophie says, "The cube root of a positive number is always smaller than the number."
Give an example to show that Sophie is wrong.
Model answer
Take 0.125. \(0.5^3 = 0.125\), so \(\sqrt[3]{0.125} = 0.5\) 0.5 is bigger than 0.125, so Sophie is wrong.
Explanation
For numbers between 0 and 1, roots are bigger than the number itself, because cubing a number between 0 and 1 makes it smaller. One counter-example is enough to show a statement is false.
Mark scheme
1 mark
M1 for choosing a number between 0 and 1 (or 1 itself) and finding its cube root
1 mark
C1 for a correct counter-example clearly shown, e.g. \(\sqrt[3]{0.125} = 0.5\) and 0.5 > 0.125
Edge length \(= \sqrt[3]{343} = 7\) cm Area of one face \(= 7^2 = 49\) cm² Total surface area = 6 × 49 = 294 cm²
Explanation
The volume of a cube is the edge length cubed, so take the cube root to find the edge. A cube has 6 identical square faces. Common errors are dividing 343 by 3, or using only one face.
Mark scheme
1 mark
P1 for \(\sqrt[3]{343} = 7\)
correct answer with no working scores full marks
1 mark
P1 for the area of one face, \(7^2 = 49\), and × 6
ft their edge length
1 mark
A1 for 294 (cm²)
cao
Question 4
4 marks
A square has an area of 40 cm². A cube has edges that are the same length as the sides of the square.
Work out an estimate for the volume of the cube. You must show your working.
Model answer
Side length \(= \sqrt{40}\) cm \(6.3^2 = 39.69\) and \(6.4^2 = 40.96\), so \(\sqrt{40} \approx 6.3\) Volume \(= (\sqrt{40})^3 = 40 \times \sqrt{40} \approx 40 \times 6.3 = 252\) cm³
Explanation
The volume is the side length cubed. Writing \((\sqrt{40})^3\) as \(\sqrt{40} \times \sqrt{40} \times \sqrt{40} = 40\sqrt{40}\) means you only need one estimate of a square root. A rough estimate of 6 for the side gives 216, which is too far from the true value of about 253.
Mark scheme
1 mark
P1 for the side length \(= \sqrt{40}\)
1 mark
P1 for estimating \(\sqrt{40}\) between 6.3 and 6.4, e.g. \(6.3^2 = 39.69\)
\(\sqrt{40} \approx 6\) scores P0 for this mark
1 mark
P1 for volume \(= (\sqrt{40})^3 = 40\sqrt{40}\), or their estimate cubed, e.g. \(6.3^3\)
1 mark
A1 for an answer in the range 245 to 260 (cm³)
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