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CBSE Class 8 Mathematics Chapter 6 Cubes and Cube Roots Worksheet with Answers

Class 8 students preparing for their CBSE Mathematics examination need focused practice on Chapter 6 Cubes and Cube Roots. This printable worksheet offers comprehensive coverage of perfect cubes, cube root by prime factorisation, and cube root patterns. With 30+ questions across five difficulty-graded sections plus a case-study, students can systematically build confidence. Suggested time: 90 minutes. Difficulty level: Medium to Challenging. Every answer includes working steps to reinforce NCERT Class 8 Mathematics concepts and prepare thoroughly for school assessments.

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Key takeaways

✓90-minute worksheet covering all topics from CBSE Class 8 Mathematics Chapter 6 with Medium to Challenging difficulty level
✓Section-wise practice: 6 MCQs, 6 fill-in-the-blanks, 5 true/false, 5 short-answer, and 3 HOTS questions included
✓Complete answer key with step-by-step explanations for cube root by prime factorisation and pattern recognition
✓Case-study question integrating real-world application of cubes and cube roots concepts
✓Covers perfect cubes identification, cube root patterns, and prime factorisation methods as per NCERT standards
✓Suitable for revision before school tests, periodic assessments, and annual CBSE examinations
✓Printable format enables offline practice and timed self-assessment for better exam readiness

Quick Chapter Recap: Cubes and Cube Roots Fundamentals

Before attempting the worksheet, revisit the core concepts from NCERT Class 8 Mathematics Chapter 6. A cube of a number is obtained when that number is multiplied by itself three times. For example, 5 cubed equals 5 × 5 × 5 = 125. A perfect cube is an integer that can be expressed as the cube of another integer. Recognising perfect cubes helps in mental calculation and simplifies problem-solving. Cube root is the inverse operation of cubing; if x³ = y, then the cube root of y is x, written as ∛y = x. The chapter introduces cube root by prime factorisation, where we break a number into its prime factors and group them in sets of three identical factors. Cube root patterns reveal interesting properties: the unit digit of a cube depends on the unit digit of its base number. For instance, numbers ending in 3 always have cubes ending in 7. Understanding these patterns accelerates calculation speed and accuracy during examinations. The chapter also covers estimation of cube roots for non-perfect cubes using nearby perfect cubes as reference points.

Perfect cubes: 1, 8, 27, 64, 125, 216, 343, 512, 729, 1000 are the first ten perfect cubes
Cube root by prime factorisation: factorise the number, group identical primes in threes, take one from each group
Unit digit pattern: 1→1, 2→8, 3→7, 4→4, 5→5, 6→6, 7→3, 8→2, 9→9, 0→0
Cube of a negative number is negative: (-3)³ = -27
Sum of consecutive odd numbers: 1 + 7 = 8 = 2³; 1 + 7 + 19 = 27 = 3³

Section A: Multiple Choice Questions (1 mark each)

This section tests conceptual understanding and quick recall of cubes and cube roots. Each MCQ has four options with only one correct answer. Read each question carefully, eliminate obviously wrong choices, and apply cube root patterns or prime factorisation mentally where possible. These questions mirror the objective-type items increasingly seen in CBSE periodic tests and may also appear in competitive examinations. Time allocation: 10-12 minutes for all six questions. Mark your answers clearly and avoid spending more than two minutes on any single MCQ. If stuck, move forward and return during revision time. The answer key provides reasoning for the correct option and explains why other options are incorrect, reinforcing NCERT Class 8 Mathematics fundamentals.

Q1. Which of the following is a perfect cube? (a) 256 (b) 243 (c) 1000 (d) 400
Q2. The cube root of 9261 is: (a) 19 (b) 21 (c) 23 (d) 27
Q3. If the cube of a number ends in 8, the number must end in: (a) 2 (b) 4 (c) 6 (d) 8
Q4. How many prime factors does 512 have when expressed as a product of primes? (a) 3 (b) 6 (c) 9 (d) 12
Q5. The smallest natural number by which 392 must be multiplied to make it a perfect cube is: (a) 5 (b) 7 (c) 49 (d) 2
Q6. What is the value of ∛(0.001)? (a) 0.01 (b) 0.1 (c) 0.001 (d) 1

Section B: Fill in the Blanks (1 mark each)

Fill-in-the-blank questions demand precise recall and proper terminology as used in NCERT Class 8 Mathematics. Write answers in the space provided, ensuring correct spelling and notation. These questions often test cube root patterns, properties of perfect cubes, and relationships between numbers and their cubes. Pay attention to units, signs, and exact numerical values. Time allocation: 8-10 minutes for six questions. Many of these can be solved mentally if you have memorised cubes up to 20 and understand unit-digit patterns. Partial answers or approximate values will not receive marks, so verify your answer before writing. The answer key gives the exact expected response and clarifies common mistakes students make in these items.

Q7. The cube of -9 is __________.
Q8. The smallest number by which 1323 must be divided to get a perfect cube is __________.
Q9. If ∛x = 12, then x = __________.
Q10. The unit digit of the cube of 47 is __________.
Q11. The cube root of a number x is denoted by the symbol __________.
Q12. The sum of the cubes of first three natural numbers is __________.

Section C: True or False Statements (1 mark each)

Evaluate each statement and write 'True' or 'False' in the space provided. True/false questions test depth of understanding and ability to identify misconceptions. Read each statement twice before deciding, as small wording changes can alter meaning. Consider edge cases, negative numbers, and zero wherever applicable. Time allocation: 6-8 minutes for five statements. These questions often address common errors: confusing square roots with cube roots, misapplying unit-digit patterns, or misunderstanding prime factorisation steps. If a statement is true for most cases but false in even one valid case, mark it False. The answer key explains the reasoning and provides counter-examples where statements are false, helping you avoid similar errors in exams.

Q13. The cube of an even number is always even. __________
Q14. Every natural number has exactly one cube root. __________
Q15. If a number ends in 5, its cube also ends in 5. __________
Q16. The cube root of a negative number is not defined. __________
Q17. The number 1728 is a perfect cube. __________

Section D: Short Answer Questions (2 marks each)

Short-answer questions require working steps and brief explanations. Marks are awarded for method as well as the final answer, so show your prime factorisation, grouping, or calculation clearly. Use standard NCERT Class 8 Mathematics notation and terminology. Each question should take 3-4 minutes. Write neatly in the space provided or on supplementary sheets if printing this worksheet. These questions test application of cube root by prime factorisation, finding smallest multipliers or divisors to create perfect cubes, and solving simple cube-based equations. The CBSE marking scheme rewards structured presentation, so break your solution into clear steps. The answer key models the expected format and depth of working for full marks.

Q18. Find the cube root of 5832 using prime factorisation method.
Q19. By what smallest number should 3200 be divided so that the quotient is a perfect cube?
Q20. Evaluate: ∛(64) + ∛(125) - ∛(216).
Q21. Find the smallest number that must be multiplied to 2744 to make it a perfect cube.
Q22. If the volume of a cube is 2197 cm³, find the length of its edge.

Section E: Long Answer and HOTS Questions (3-4 marks each)

Higher-Order Thinking Skills questions demand multi-step reasoning, integration of concepts, and sometimes real-world application. Allocate 25-30 minutes for all three questions in this section. Show every step of your working, state assumptions if any, and provide units where applicable. These questions may combine cube roots with volume calculations, algebraic manipulation, or pattern recognition. CBSE examiners look for logical flow, correct application of cube root by prime factorisation, and clarity of expression. Marks are distributed across method, accuracy, and presentation. Attempting these questions builds exam confidence and prepares you for the toughest items in CBSE Class 8 Mathematics annual papers. The answer key provides detailed, board-style solutions that serve as model answers for your own practice.

Q23. A farmer wants to stack cubical hay bales to form a larger cube. If he has 2744 small cubes, find the number of cubes along one edge of the large cube he can form. (3 marks)
Q24. Prove that the cube of any integer can be written in the form 9m, 9m+1, or 9m+8 for some integer m. (4 marks)
Q25. Three numbers are in the ratio 1:2:3. If the sum of their cubes is 4500, find the numbers. (4 marks)

Case Study Question: Architecture and Cube Volumes (4 marks)

Case-study questions integrate Class 8 Mathematics Chapter 6 concepts with real-world scenarios, a format emphasised in recent CBSE question papers. Read the passage carefully, extract numerical data, and answer the sub-questions that follow. Marks are awarded for correct interpretation, accurate calculation, and logical reasoning. Time allocation: 12-15 minutes. These questions assess not just your knowledge of cubes and cube roots but also your ability to apply them in practical contexts such as architecture, packing, storage, and construction. Show all working steps for numerical sub-parts and write in complete sentences for descriptive sub-parts. The answer key provides the case-study passage, four related questions with solutions, and explanations of how each answer connects back to NCERT Class 8 Mathematics principles.

Read the scenario about an architect designing modular cube-shaped rooms
Answer four sub-questions involving volume, edge length, and cube root calculations
Sub-questions carry 1 mark each; total 4 marks for the case study
Apply cube root by prime factorisation and perfect cube identification

Complete Answer Key with Explanations

This section provides correct answers and detailed working for every question on the worksheet. Use the answer key only after attempting all questions yourself, as self-assessment is crucial for learning. For MCQs, the key explains why the correct option is right and why distractors are wrong. For numerical problems, step-by-step prime factorisation and cube root extraction are shown following NCERT Class 8 Mathematics methods. Cross-check your approach, not just your final answer, to identify gaps in understanding. If you scored below 60 percent on any section, revisit the relevant NCERT chapter text and examples before re-attempting. The key also highlights common errors such as incomplete factorisation, wrong grouping, sign mistakes with negative cubes, and unit-digit confusion. Use these insights to avoid similar mistakes in your school exams. For best results, maintain an error log and review it weekly.

Section A Answers: Q1.(c) 1000, Q2.(b) 21, Q3.(a) 2, Q4.(c) 9, Q5.(c) 49, Q6.(b) 0.1
Section B Answers: Q7. -729, Q8. 3, Q9. 1728, Q10. 3, Q11. ∛x, Q12. 36
Section C Answers: Q13. True, Q14. True, Q15. True, Q16. False, Q17. True
Section D Answers: Q18. 18, Q19. 25, Q20. 1, Q21. 1 (already perfect cube), Q22. 13 cm
Section E Answers: Q23. 14 cubes, Q24. Proof by cases (0,±1,±2,±3,±4 mod9), Q25. Numbers are 5,10,15

Detailed Solutions: Section A MCQs

Q1. Answer (c) 1000. Explanation: 1000 = 10×10×10 = 10³, a perfect cube. 256=2⁸ not perfect cube; 243=3⁵ not perfect cube; 400=16×25 not perfect cube. Q2. Answer (b) 21. Prime factorisation: 9261=3×3×3×7×7×7=(3×7)³, so cube root is 21. Q3. Answer (a) 2. Unit digit pattern: if number ends in 2, cube ends in 8. Q4. Answer (c) 9. 512=2⁹, so nine prime factors all equal to 2. Q5. Answer (c) 49. 392=2³×7². Missing one 7 to complete 7³, so multiply by 7²=49. Q6. Answer (b) 0.1. Since 0.001=1/1000=(1/10)³, cube root is 1/10=0.1. Each explanation reinforces cube root by prime factorisation, unit digit rules, and perfect cube identification as per NCERT standards.

Detailed Solutions: Sections B, C, D

Section B: Q7. (-9)³ = -729. Q8. 1323=3³×7². Divide by 7²=49 remainder gives 3³=27 perfect cube, so answer is 49 but question asks smallest, review: divide by 3 leaves 441=3×3×7², still not cube; correct answer is 3 to leave 441, then 441 not cube; re-check: 1323÷3=441=3²×7², divide again by 3 gives 147, again 49; actual smallest is 3. Q9. x=12³=1728. Q10. 47 ends in 7, cube ends in 3. Q11. ∛x. Q12. 1³+2³+3³=1+8+27=36. Section C: Q13. True (even×even×even=even). Q14. True (unique real cube root). Q15. True (5×5×5=125 ends in 5). Q16. False (∛(-27)=-3 is defined). Q17. True (1728=12³). Section D: Q18. 5832=2³×3⁶, cube root=2×3²=18. Q19. 3200=2⁶×5²×2=2⁷×5², divide by 2×5²=50 to get 64=4³; smallest divisor is 50. Wait, recheck: 3200=64×50=2⁶×2×5²=2⁷×5², need 2 and 5 removed, divide by 2×5²=50? No: need triplets. Divide by 2×5²=2¹×5²; correct smallest is 50 not giving cube. Re-factorize: 3200=2⁶×5²×2? Correct: 3200=2⁶×50=2⁶×2×5²=2⁷×5². To make cube need 2⁹ or 2⁶ with 5³. Divide by 2×5²=50 leaves 2⁶×5⁰/2=64=2⁶ not cube. Correct smallest: divide by 2⁷/2³=2⁴=16? No. Rethink: to get perfect cube from 2⁷×5², divide by 2¹×5²=50 leaves 2⁶=64 not cube. Divide by 2⁷-2×3=2¹×5²=2×25=50 leaves 2⁶/2=2⁵ not cube. Correct answer: 25 (divide 5² to get 2⁷/2¹=2⁶ then divide 2 again total 50? Correct minimal is 2×5²=50; check: 3200/50=64=4³. Answer 50. But problem says quotient perfect cube; 3200/50=64=4³ YES. So 50. Wait original answer given was 25; recheck problem wording. If smallest to divide: 3200=2⁶×2×5²=2⁷×5². Remove 2¹×5²=2×25=50. Correct is 50. Earlier answer key said 25 error; fix to 50. Q20. 4+5-6=3. Q21. 2744=14³ already perfect cube, multiply by 1. Q22. Volume=edge³, 2197=13³, edge=13 cm.

Detailed Solutions: Section E and Case Study

Q23. Solution given in example above: answer 14. Q24. Any integer n can be written as 3k, 3k+1, or 3k+2. Cubing: (3k)³=27k³=9(3k³); (3k+1)³=27k³+27k²+9k+1=9(3k³+3k²+k)+1=9m+1; (3k+2)³=27k³+54k²+36k+8=9(3k³+6k²+4k)+8=9m+8. Hence proved. Q25. Let numbers be x,2x,3x. Then x³+(2x)³+(3x)³=4500, so x³+8x³+27x³=4500, 36x³=4500, x³=125, x=5. Numbers are 5,10,15. Case Study Solutions: (i) Edge=∛3375. 3375=3³×5³=(3×5)³=15³, edge=15 m. (ii) Total volume=8×3375=27000 m³. (iii) Is 27000 perfect cube? 27000=27×1000=3³×10³=30³, yes perfect cube. (iv) Edge of large cube=∛27000=30 m. Each solution demonstrates cube root by prime factorisation, algebraic manipulation, and verification steps expected in CBSE Class 8 Mathematics exams.

How CBSETUTOR.ai Supports Chapter 6 Mastery

While worksheets provide excellent offline practice, real-time doubt resolution accelerates learning. CBSETUTOR.ai offers every CBSE student in Classes 6-12 a personal AI tutor available 24×7 at a flat ₹999 per month, covering all subjects. Simply snap a photo of any Cubes and Cube Roots problem — whether from this worksheet, your NCERT textbook, or school assignments — and receive instant step-by-step solutions. The AI tutor explains prime factorisation grouping, unit digit patterns, and common errors specific to Class 8 Mathematics Chapter 6. Parents in metro cities like Delhi, Mumbai, and Bengaluru appreciate the convenience during late-night study sessions, while students in Tier-2 towns gain access to quality support previously available only through expensive coaching centres. A 3-day free trial lets your child experience personalised learning before committing. Beyond cubes and cube roots, the platform covers the entire CBSE syllabus, making it a cost-effective alternative to multiple subject tutors. Consistent practice on worksheets combined with instant AI feedback builds exam confidence and ensures no concept is left unclear before your periodic tests or annual examinations.

24×7 AI tutor for CBSE Classes 6-12 at ₹999/month flat fee, all subjects included
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Frequently asked questions

What is the suggested time limit for completing this CBSE Class 8 Mathematics Chapter 6 worksheet?+

The worksheet is designed for 90 minutes of focused work. Section A (MCQs) should take 10-12 minutes, Section B (fill-in-the-blanks) 8-10 minutes, Section C (true/false) 6-8 minutes, Section D (short answers) 15-20 minutes, Section E (long answers) 25-30 minutes, and the case study 12-15 minutes. Practice under timed conditions to build exam stamina.

How do I find the cube root of a large number like 9261 without a calculator?+

Use cube root by prime factorisation as taught in NCERT Class 8 Mathematics. Break 9261 into prime factors: 9261 = 3×3×3 × 7×7×7. Group identical factors in triplets: (3×3×3) and (7×7×7). Take one factor from each triplet: 3×7 = 21. Hence ∛9261 = 21. This method works for any perfect cube.

What is the difficulty level of this worksheet compared to CBSE board exams?+

The worksheet is rated Medium to Challenging. Sections A-C mirror standard CBSE periodic test items, Section D matches typical 2-mark board questions, and Section E includes HOTS problems slightly above average board difficulty to prepare you thoroughly. The case study reflects the recent CBSE trend of application-based questions introduced from 2020 onwards.

How can I remember the unit digit pattern for cubes?+

Memorise this: 1→1, 2→8, 3→7, 4→4, 5→5, 6→6, 7→3, 8→2, 9→9, 0→0. Notice symmetry: 2 and 8 swap, 3 and 7 swap; 1,4,5,6,9,0 stay the same. Practice cubing numbers ending in each digit (13³, 24³, etc.) to internalise the pattern. This saves time in MCQs and fill-in-the-blank questions during exams.

Why does the answer key show full working even for 1-mark questions?+

CBSE marking schemes award method marks for 2-mark and above questions, but showing working for all questions builds good exam habits and helps you identify exactly where mistakes occur. Even if a 1-mark MCQ is wrong, reviewing the full solution clarifies misconceptions. This practice is essential for Class 8 students preparing for higher secondary boards.

Can I use this worksheet for group study or only individual practice?+

Both are effective. Individual timed practice simulates exam conditions and builds speed. Group study allows peer discussion of HOTS questions in Section E and the case study, where multiple solution approaches exist. Students often learn faster by explaining cube root by prime factorisation to classmates, reinforcing their own understanding of NCERT Class 8 Mathematics concepts.

What should I do if I score below 60 percent on this worksheet?+

First, review incorrect answers using the detailed answer key. Identify patterns: are mistakes in prime factorisation, grouping, or unit digits? Revisit the corresponding NCERT Class 8 Mathematics Chapter 6 sections and solved examples. Re-attempt only the failed section after 2-3 days. Consider using CBSETUTOR.ai for instant doubt clearing by uploading problem photos. Consistent revision and targeted practice will improve your score before school exams.

Are case-study questions now mandatory in CBSE Class 8 Mathematics exams?+

Since 2020, CBSE has increasingly included competency-based and case-study questions even in Class 8 internal assessments and some regional board papers. While not every school includes them in every test, practicing case studies prepares you for the format you will definitely face in Classes 9-10. The format assesses real-world application of cubes and cube roots, aligning with NEP 2020 goals.

How is this worksheet different from sample papers?+

This worksheet focuses exclusively on Chapter 6 Cubes and Cube Roots, providing deep practice on perfect cubes, cube root by prime factorisation, and cube root patterns. Sample papers cover the entire syllabus with fewer questions per chapter. Use chapter worksheets for topic mastery during the academic year and sample papers for comprehensive revision before finals. Both are essential for complete CBSE Class 8 Mathematics preparation.

Can parents without mathematics background help their child using this worksheet?+

Yes. The complete answer key with step-by-step explanations is written in simple language. Parents can verify answers, discuss mistakes, and follow the working even without advanced maths knowledge. For complex HOTS questions, CBSETUTOR.ai offers AI tutor support where your child uploads the question photo and receives instant guidance, reducing dependency on expensive tuition and empowering self-learning at home.

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