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Harder indices

Free Printable Harder indices PDF Worksheets, Revision Guide & GCSE Past Paper Questions (Grade 6)

Download free harder indices worksheets (PDF), GCSE maths past paper questions and worked solutions for Harder indices (6B). This is a Grade 6 number topic from the GCSE questions by topic collection for Higher tier 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, Maths4Everyone, Corbett Maths, Dr Frost, 1st Class Maths, Metatutor, Maths Takeaway, Maths DIY, Dr Austin, E Classroom and PMT, along with past paper question booklets.

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

Free Harder indices PDF Worksheets & Answers

Grade 6 • Topic 6B
Resource Questions Answers
GCSE past paper questions & solutions
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Graded exam questions & solutions
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GCSE 9–1 practice booklet & solutions
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Textbook Exercise & answers
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GCSE Maths Takeaway exam questions
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PMT (Edxl)
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PMT (AQA)
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PMT (IGCSE)
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Grade 6 topics
6A - Recurring decimals6B - Harder indices
6C - The product rule6D - Repeated change
6E - Triple brackets6F - Parallel and perpendicular
6G - Inequalities on graphs6H - Similar shapes (area and vol)
6J - Negative scale factors6K - Circle theorems
6L - Cumulative frequency6M - Box plots
6N - Combined transformations6P - Capture recapture
GCSE Maths Grade 6 Revision Notes & Method

Guide to Harder indices: Explanation, Diagrams & Worked Examples

Download Printable One Page Guide (PDF)

Explanation:

For a fractional index mn, the bottom number n is the root and the top number m is the power: am/n = (n√a)m. Always take the root first to keep the numbers small before raising to the power.

A negative index means take the reciprocal (flip the fraction upside down): a−n = 1an and (ab)−n = (ba)n.

Example 1: Find the exact value of (a) 272/3 and (b) 16−3/4.

Solution:

(a) Cube root first, then square: (3√27)2 = 32 = 9
(b) Negative power gives reciprocal, 4th root is 2:
16−3/4 = 1(4√16)3 = 123 = 18
Answer: (a) 9, (b) 18

Example 2: Work out the value of (8125)−2/3 as an improper fraction.

Solution:

Negative sign flips the fraction: (1258)2/3
Cube root top and bottom: (3√1253√8)2 = (52)2
Square top and bottom: 52 ÷ 22 = 254
Answer: 254 (or 614)
Scanned GCSE Exam Questions & Worked Solutions

GCSE Past Paper Harder indices Questions & Answers

Past Paper Example 1 GCSE Nov 2017 1H Q10 3 marks
Full Paper (PDF) • Mark Scheme (PDF)
Work out
(a) Write down the value of 100 {}1/2) (1) (b) Find the value of 125 {}2/3) (2)
(a) Write down the value of 100<span class="opg-frac"><span class="opg-num">1</span><span class="opg-den">2</span></span> [1 mark]
(b) Find the value of 125<span class="opg-frac"><span class="opg-num">2</span><span class="opg-den">3</span></span> [2 marks]
Worked Solution & Mark Scheme (3 marks)
Part (a) Solution:
√100 = 10 B1
Answer (a): 10
Part (b) Solution:
(∛125)² = 5² = 25 M1 A1
Answer (b): 25
Past Paper Example 2 GCSE Nov 2024 1H Q10 4 marks
Full Paper (PDF) • Mark Scheme (PDF)
Work out
(a) Work out 251/2 × 81/3 (2)
(b) Find the value of (1/32)3/5 (2)
(a) Work out 25<span class="opg-frac"><span class="opg-num">1</span><span class="opg-den">2</span></span> × 8<span class="opg-frac"><span class="opg-num">1</span><span class="opg-den">3</span></span> [2 marks]
(b) Find the value of (132)<span class="opg-frac"><span class="opg-num">3</span><span class="opg-den">5</span></span> [2 marks]
Worked Solution & Mark Scheme (4 marks)
Part (a) Solution:
5 × 2 = 10 M1 A1
Answer (a): 10
Part (b) Solution:
(1/2)³ = 1/8 M1 A1
Answer (b): 18
Past Paper Example 3 GCSE May 2023 1H Q14 4 marks
Full Paper (PDF) • Mark Scheme (PDF)
(a) Write 1/16 in the form 4ⁿ. (1) (b) Work out the value of 8 {}5/3) – 9 {}3/2). (3)
(a) Write 116 in the form 4n where n is an integer. [1 mark]
(b) Work out the value of 8<span class="opg-frac"><span class="opg-num">5</span><span class="opg-den">3</span></span> - 9<span class="opg-frac"><span class="opg-num">3</span><span class="opg-den">2</span></span> [3 marks]
Worked Solution & Mark Scheme (4 marks)
Part (a) Solution:
1/16 = 1/4² = 4⁻² B1
Answer (a): 4-2
Part (b) Solution:
(³√8)⁵ – (√9)³ = 2⁵ – 3³ = 32 – 27 = 5 M2 A1
Answer (b): 5

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