Question Bank › GCSE Algebra › Indices
Indices Topic Simplifying Basic Expressions (17) Expanding Brackets (13) Substitution (10) Drawing & Using Graphs (34) Solving Simple Equations (21) Factorising (9) Algebraic Fractions (4) Linear Graphs & Gradients (26) Forming Equations (15) Solving Quadratics (10) Quadratic Inequalities (2) Linear Inequalities (9) Simultaneous Equations (9) Completing the Square (2) Functions (9) Graphical Transformations (2) Indices (14) Manipulating Formulae/ Changing the Subject (9) Sequences (19) Iterations (1) Current PowerPoint version
All boards Edexcel AQA All specs Current spec All series June 2025 November 2024 November 2023 June 2017 Any marks 1 to 4 marks 5 to 8 marks 9+ marks
Questions Step through List
‹ Previous All questions Next ›
Foundation June 2025 Paper 3 Q21 21
(a) Simplify \(y^{10} \div y^2\) (1)
(b) Simplify \((e^3)^2\) (1)
(c) Expand \(x(3x^2 + 5)\) (1)
(d) Expand and simplify \((n + 5)(n - 2)\) (2)
Mark scheme (a) Mark scheme (b) Mark scheme (c) Mark scheme (d)
Mark scheme (a) Answer Mark Mark scheme \(y^8\) B1 cao
Mark scheme (b) Answer Mark Mark scheme \(e^6\) B1 cao
Mark scheme (c) Answer Mark Mark scheme \(3x^3 + 5x\) B1 cao
Mark scheme (d) Answer Mark Mark scheme \(n^2 + 3n - 10\) M1 for 3 out of no more than 4 terms correct with correct signs or 4 correct terms ignoring signs A1 cao
Additional guidance \(n^2 - 2n + 5n - 10\)
Higher June 2025 Paper 2 Q17 17 Here is a sketch of part of the graph of \(y = k^x\) where \(k\) is a positive constant.
The graph passes through the point with coordinates \((-1, 1.5)\)
Find the value of \(k\). (2)
Mark scheme
Mark scheme Answer Mark Mark scheme \(\dfrac{2}{3}\) M1 for correct substitution, eg \(1.5 = k^{-1}\) oe or \(\dfrac{1}{1.5}\) A1 oe but not \(\dfrac{1}{1.5}\)
Additional guidance Accept rounded or truncated to 2dp or better
Higher June 2025 Paper 3 Q11 11 \(\dfrac{3^n \times 3^{20}}{3^8} = 3^{14}\)
(a) Find the value of \(n\). (2)
(b) Write \(\left(64m^{12}\right)^{\frac{2}{3}}\) in the form \(am^b\) where \(a\) and \(b\) are integers. (2)
Mark scheme (a) Mark scheme (b)
Mark scheme (a) Answer Mark Mark scheme 2 M1 for a correct start, using one rule of indices,or for forming an equation in \(n\), A1 cao
Additional guidance Allow an answer of \(3^n = 3^2\)
An answer of 9 or \(3^n = 9\), \(n \neq 2\) on its own is to be awarded 0 marks
Mark scheme (b) Answer Mark Mark scheme \(16m^8\) M1 for an intention to find the cube root and square, A1 cao
Additional guidance Do not condone missing brackets
16 or \(m^8\) imply M1
Accept \(a = 16\), \(b = 8\)
Higher June 2025 Paper 2 Q9 9 \(3x^{-1}(4x - x^3) = a + bx^n\) for all the values of \(x\) that are not zero.
Find the value of \(a\), the value of \(b\) and the value of \(n\). (2)
Mark scheme
Mark scheme Answer Mark Mark scheme \(12, -3, 2\) B2 all correct (B1 for one or two correct)
Additional guidance \(12 - 3x^2\) seen in working space gets B2 unless contradicted
May be seen in an expression of the correct form, eg \(a + bx^n\)
Foundation November 2024 Paper 3 Q25 25
(a) Simplify fully \(2x^3y^5 \times 7x^2y\) (2)
(b) Simplify \((m^2)^{-3}\) (1)
Mark scheme (a) Mark scheme (b)
Mark scheme (a) Answer Mark Mark scheme \(14x^5y^6\) B2 cao (B1 for correct simplification of two termsor \(14x^by^6\) or \(14x^5y^c\) where \(a \neq 14\), \(b \neq 5\), \(c \neq 6\))
Additional guidance Where \(a\), \(x^b\), \(y^c\) can be made up of two products
Mark scheme (b) Answer Mark Mark scheme \(m^{-6}\) B1 for \(m^{-6}\) or \(\dfrac{1}{m^6}\)
Higher November 2024 Paper 1 Q10 10
(a) Work out \(25^{\frac{1}{2}} \times 8^{\frac{1}{3}}\) (2)
(b) Find the value of \(\left(\dfrac{1}{32}\right)^{\frac{3}{5}}\) (2)
Mark scheme (a) Mark scheme (b)
Mark scheme (a) Answer Mark Mark scheme 10 M1 for \(25^{\frac{1}{2}} = 5\) or \(8^{\frac{1}{3}} = 2\) A1 cao
Additional guidance Accept \(25^{\frac{1}{2}} = -5\) for M1 only
Mark scheme (b) Answer Mark Mark scheme \(\dfrac{1}{8}\) M1 for \(\left(\dfrac{1}{\sqrt[5]{32}}\right)^3\) or \(\left(\dfrac{1}{2}\right)^3\) or \(\sqrt[5]{\dfrac{1}{32^3}}\) or \(\sqrt[5]{\dfrac{1}{32768}}\) A1 for \(\dfrac{1}{8}\) oe
Higher November 2024 Paper 3 Q9 9
(a) Simplify fully \(2x^3y^5 \times 7x^2y\) (2)
(b) Simplify \((m^2)^{-3}\) (1)
Mark scheme (a) Mark scheme (b)
Mark scheme (a) Answer Mark Mark scheme \(14x^5y^6\) B2 cao (B1 for correct simplification of two termsor \(14x^by^6\) or \(14x^5y^c\) where \(a \neq 14\), \(b \neq 5\), \(c \neq 6\))
Additional guidance Where \(a\), \(x^b\), \(y^c\) can be made up of two products
Mark scheme (b) Answer Mark Mark scheme \(m^{-6}\) B1 for \(m^{-6}\) or \(\dfrac{1}{m^6}\)
Foundation November 2024 Paper 2 Q25 25 Here are three terms.
\(xy\) \(x^2\) \(5y^2\)
Alec multiplies two of these terms.
Work out the three possible fully simplified answers. [3 marks]
Mark scheme
Mark scheme Answer Mark Comments \(x^3y\) or \(yx^3\) B1 \(5xy^3\) or \(5y^3x\) B1 \(5x^2y^2\) or \(5y^2x^2\) B1
Additional guidance Mark the answer lines unless blank
Do not allow transcription errors
Foundation November 2023 Paper 1 Q22 22 Work out the value of \(\quad (8^2 \times 8) \div (8^9 \div 8^5)\)
Give your answer as a decimal. [3 marks]
Mark scheme
Mark scheme Answer Mark Comments \((8^2 \times 8 =)\ 8^3\) or \((8^9 \div 8^5 =)\ 8^4\) M1 \((8^3\) or \(512) \div (8^4\) or \(4096)\) M1dep oe in the form \(8^n \div 8^{(n + 1)}\) (0).125 A1
Additional guidance (0).125 and either \(8^{-1}\) or \(\dfrac{1}{8}\) on the answer line M1M1A1 (0).125 in working and \(8^{-1}\) on the answer line M1M1A0 If a student attempts numerical and index working award the higher mark
Higher June 2017 Paper 1 Q24 24
(a) Work out \(\quad \sqrt{12\tfrac{1}{4}} \quad\) as an improper fraction. [1 mark]
(b) Work out \(\quad \sqrt[3]{16} \quad\) as a power of 2 [2 marks]
Mark scheme (a) Mark scheme (b)
Mark scheme (a) Answer Mark Comments \(\dfrac{7}{2}\) B1 oe improper fraction
Additional guidance Condone \(\pm\) on numerator and/or denominator
Mark scheme (b) Answer Mark Comments \((16 =)\ 2^4\) or \((\sqrt[3]{16} =)\ 16^{\frac{1}{3}}\) or \(\sqrt[4]{16} = 2\) M1 oe \(2^{\frac{4}{3}}\) or \(2^{1\frac{1}{3}}\) or \(2^{1.\dot{3}}\) A1
Additional guidance \(\sqrt[3]{16} = 2^4\) not recovered M0A0
Foundation June 2017 Paper 1 Q17 17 Circle the expression which does not simplify to \(\;y^3\) [1 mark]
\(y \times y \times y\) \(y^4 \div y\) \(y^2 \times y\) \(y^6 \div y^2\) Mark scheme
Mark scheme Answer Mark Comments \(y^6 \div y^2\) B1
Higher June 2017 Paper 3 Q12 12 \(\left(ar^b\right)^4 = 16r^{20} \quad\) where \(a\) and \(b\) are positive integers.
Work out \(a\) and \(b\) [2 marks]
Mark scheme
Mark scheme Answer Mark Comments \(a = 2\) B1 May be embedded \(b = 5\) B1 May be embedded
Additional guidance \((2r^5)^4\) B1B1 \((r^5)^4\) B1 \(2^4 = 16\) on its own is not enough B0 \(a = 5\) and \(b = 2\) B0B0
Foundation June 2017 Paper 3 Q4 4 Circle the value of \(2^5\) [1 mark]
Mark scheme
Mark scheme Higher June 2017 Paper 1 Q1 1 Simplify \(\quad 2^5 \times 2^3\)
Circle your answer. [1 mark]
\(4^8\) \(2^8\) \(2^{15}\) \(4^{15}\) Mark scheme
Mark scheme Answer Mark Comments \(2^8\) B1
No questions match these filters.
