Integer and fractional indices - Edexcel GCSE Maths (Calculator) Exam Questions

Practise 28 exam-style questions on Integer and fractional indices for Edexcel GCSE Maths (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

Use your calculator to work out
\[\frac{\sqrt{6.4^2 + 3.85}}{1.7^3 - 2.06}\]
Give your answer correct to 3 significant figures.

Model answer

Numerator: \(\sqrt{6.4^2 + 3.85} = \sqrt{40.96 + 3.85} = \sqrt{44.81} = 6.694027...\)
Denominator: \(1.7^3 - 2.06 = 4.913 - 2.06 = 2.853\)
\(6.694027... \div 2.853 = 2.346311...\)
= 2.35 (3 s.f.)

Explanation

Work out the numerator and the denominator separately, or use the fraction key with brackets, and only round at the very end. A common mistake is typing the expression in one line without brackets, so the calculator only divides part of the numerator by part of the denominator.

Mark scheme

1 mark
M1 for \(\sqrt{44.81}\) or 6.694... or for 2.853 seen
numerator or denominator correctly evaluated
1 mark
A1 for 2.35
awrt 2.35 (2.3463...); correct answer with no working scores full marks

Question 2

2 marks

Write \(\dfrac{\sqrt{125}}{\sqrt[3]{25}}\) in the form \(5^{k}\), where k is a fraction.

Model answer

\(\sqrt{125} = \left(5^{3}\right)^{\frac{1}{2}} = 5^{\frac{3}{2}}\)
\(\sqrt[3]{25} = \left(5^{2}\right)^{\frac{1}{3}} = 5^{\frac{2}{3}}\)
\(5^{\frac{3}{2} - \frac{2}{3}} = 5^{\frac{9}{6} - \frac{4}{6}} = 5^{\frac{5}{6}}\)

Explanation

Write 125 and 25 as powers of 5, then use roots as fractional powers: a square root multiplies the index by \(\frac{1}{2}\) and a cube root by \(\frac{1}{3}\). Subtract the indices because you are dividing, using a common denominator of 6.

Mark scheme

1 mark
M1 for \(\sqrt{125} = 5^{\frac{3}{2}}\) or \(\sqrt[3]{25} = 5^{\frac{2}{3}}\)
correct answer with no working scores full marks
1 mark
A1 for \(5^{\frac{5}{6}}\)
oe, accept \(k = \frac{5}{6}\)

Question 3

3 marks

The time, T seconds, for one complete swing of a pendulum of length L metres is given by
\[T = 2\pi\sqrt{\frac{L}{9.81}}\]
A clock pendulum takes 2.4 seconds for one complete swing.

Work out the length of the pendulum.
Give your answer correct to 2 decimal places.

Model answer

\(2.4 = 2\pi\sqrt{\dfrac{L}{9.81}}\)
\(\sqrt{\dfrac{L}{9.81}} = \dfrac{2.4}{2\pi} = 0.381971...\)
\(\dfrac{L}{9.81} = 0.381971...^2 = 0.145902...\)
\(L = 9.81 \times 0.145902... = 1.43130...\)
L = 1.43 m (2 d.p.)

Explanation

Undo the operations in reverse order: divide by \(2\pi\), square to remove the square root, then multiply by 9.81. Keep the unrounded value at each step. A common mistake is to square 2.4 but forget to square the \(2\pi\) as well.

Mark scheme

1 mark
M1 for \(\sqrt{\dfrac{L}{9.81}} = \dfrac{2.4}{2\pi}\) (= 0.3819...)
oe
1 mark
M1 for \(L = 9.81 \times \left(\dfrac{2.4}{2\pi}\right)^2\)
oe, squaring and multiplying by 9.81 in a correct order
1 mark
A1 for 1.43 m
awrt 1.43 (1.4313...); correct answer with no working scores full marks

Question 4

4 marks

Solve
\[\sqrt{8^{x}} = \frac{4^{x-3}}{\sqrt{2}}\]

Model answer

Left-hand side: \(\sqrt{8^{x}} = \left(2^{3x}\right)^{\frac{1}{2}} = 2^{\frac{3x}{2}}\)
Right-hand side: \(\dfrac{4^{x-3}}{\sqrt{2}} = \dfrac{2^{2x-6}}{2^{\frac{1}{2}}} = 2^{2x - \frac{13}{2}}\)
\(\frac{3x}{2} = 2x - \frac{13}{2}\)
\(3x = 4x - 13\)
\(x = 13\)

Explanation

Write everything as a power of 2: \(8 = 2^3\), \(4 = 2^2\) and \(\sqrt{2} = 2^{\frac{1}{2}}\). A square root halves an index, and dividing subtracts indices. Then equate the indices and solve; multiplying through by 2 clears the fractions. Remember to multiply the whole bracket: \(4^{x-3} = 2^{2x-6}\), not \(2^{2x-3}\).

Mark scheme

1 mark
M1 for the left-hand side as \(2^{\frac{3x}{2}}\)
oe; correct answer with no working scores full marks
1 mark
M1 for the right-hand side as \(2^{2x - 6 - \frac{1}{2}}\)
oe, e.g. \(2^{2(x-3)} \div 2^{\frac{1}{2}}\) or \(2^{2x - \frac{13}{2}}\)
1 mark
M1 for equating their indices, e.g. \(\frac{3x}{2} = 2x - \frac{13}{2}\)
both sides must be powers of 2
1 mark
A1 for \(x = 13\)
cao

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