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Indices (index laws)

Free Printable Indices (index laws) PDF Worksheets, Revision Guide & GCSE Past Paper Questions (Grade 4)

Download free indices (index laws) worksheets (PDF), GCSE maths past paper questions and worked solutions for Indices (index laws) (4B). This is a Grade 4 number topic from the GCSE questions by topic collection for Foundation and Higher tier crossover 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 Made Easy, Dr Frost, Naiker Maths, Hannah Kettle, Dr Austin, Metatutor, GCSE Maths Rocket, E Classroom and PMT, along with past paper question booklets.

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

Free Indices (index laws) PDF Worksheets & Answers

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GCSE Maths Grade 4 Revision Notes & Method

Guide to Indices (index laws): Explanation, Diagrams & Worked Examples

Download Printable One Page Guide (PDF)

Explanation:

Learn the four core index laws for powers with the same base: (1) Multiply → add powers: am × an = am+n;   (2) Divide → subtract powers: am ÷ an = am−n;   (3) Brackets → multiply powers: (am)n = amn;   (4) Power of zero: a0 = 1.

When an expression includes normal numbers (coefficients) in front of the letters, treat the numbers as normal numbers (multiply, divide, or raise them to the power) and apply the index laws to the powers!

Example 1: Simplify:
(a) y7 × y4y3      (b) (2x3y5)4

Solution:

(a) Top: y7+4 = y11; then y11−3 = y8
(b) Raise 2 to power 4: 24 = 16
Multiply powers: x3×4y5×4 = x12y20
Answer: (a) y8   (b) 16x12y20

Example 2: Simplify 18a6b23a2b5 and find the value of n in 5n × 53 = 511.

Solution:

Numbers: 18 ÷ 3 = 6; Powers: a6−2 = a4, b2−5 = b−3
For 5n × 53 = 511: n + 3 = 11 → n = 8
Answer: 6a4b−3 (or 6a4b3) and n = 8
Scanned GCSE Exam Questions & Worked Solutions

GCSE Past Paper Indices (index laws) Questions & Answers

Past Paper Example 1 Eduqas Jun 2024 2F Q17 / 2H Q2 5 marks
(a) Factorise x2 - 5x - 24.
(b) Simplify (3h2)3.
(c) Represent the inequality -2 < x ≤ 3 on the number line below.
(a) Factorise x2 - 5x - 24. [2 marks]
(b) Simplify (3h2)3. [2 marks]
(c) Represent the inequality -2 < x ≤ 3 on the number line below. [1 mark]
x–4–3–2–101234
Worked Solution & Mark Scheme (5 marks)
Part (a) Solution:
Two numbers that multiply to -24 and add to -5 are -8 and 3: (x - 8)(x + 3). M1 A1
Answer (a): (x - 8)(x + 3)
Part (b) Solution:
33 × (h2)3 = 27h^{2 × 3} = 27h6. M1 A1
Answer (b): 27h6
Part (c) Solution:
< requires an empty circle at -2
≤ requires a solid circle at 3
line connects them. B1
x–4–3–2–101234
Answer (c): Open circle at -2, filled circle at 3, connected with a line
Past Paper Example 2 GCSE Nov 2024 3H Q9 3 marks
(a) Simplify fully 2x³y⁵ × 7x²y (2)
(b) Simplify (m²)⁻³ (1)
(a) Simplify fully 2x3y5 × 7x2y [2 marks]
(b) Simplify (m2)^{-3} [1 mark]
Worked Solution & Mark Scheme (3 marks)
Part (a) Solution:
2×7 = 14, x³⁺² = x⁵, y⁵⁺¹ = y⁶ M1 A1
Answer (a): 14x5y6
Part (b) Solution:
m²×⁻³ = m⁻⁶ B1
Answer (b): m-6
Past Paper Example 3 IGCSE Jun 2025 1F Q21 / 1H Q4 4 marks
(a) x⁵ × x⁷ = xᵐ, find m. (1) (b) y⁸ ÷ y³ = yⁿ, find n. (1) (c) Simplify fully (5a⁴r²)³. (2)
(a) Find the value of m [1 mark]
(b) Find the value of n [1 mark]
(c) Simplify fully (5a4r2)3 [2 marks]
Worked Solution & Mark Scheme (4 marks)
Part (a) Solution:
5 + 7 = 12 B1
Answer (a): m = 12
Part (b) Solution:
8 – 3 = 5 B1
Answer (b): n = 5
Part (c) Solution:
5³ a {}4×3) r {}2×3) = 125a¹²r⁶ M1 A1
Answer (c): 125a12r6

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