Class 9 Mathematics Chapter 6: Cubes and Cube Roots – Complete Important Questions with Solutions

The CBSE Class 9 Mathematics board exam emphasizes conceptual clarity and application over rote memorization. Chapter 6 (Cubes and Cube Roots) typically contributes 4–6 marks to the final paper, spread across 1-mark MCQs, 2-mark short-answer questions, and occasionally 3-mark case-study or problem-solving items. The revised curriculum focuses on three key competencies: (1) identifying and constructing perfect cubes; (2) finding cube roots efficiently using prime factorisation; (3) recognizing cube root patterns and using them to solve real-world problems. Recent board papers have shifted away from straightforward computation toward reasoning-based questions. For example, instead of asking 'Find ³√1728', examiners now ask 'If the volume of a cube is 512 cm³, find the length of its side' or 'Show that 2³ × 3³ is a perfect cube.' This guide prioritizes pattern-based, conceptual questions that train you to think like an examiner. By practicing these 18 questions systematically, you'll develop the flexibility to handle unexpected question variations—a critical skill for securing high marks in CBSE exams.

Multiple-choice questions in Class 9 Cubes and Cube Roots test quick recall of definitions, basic properties, and mental computation. Here are 5 representative 1-mark questions: **Q1.** Which of the following is a perfect cube? (a) 125 (b) 130 (c) 135 (d) 140 **Answer: (a) 125** Explanation: 125 = 5³. Verify: 5 × 5 × 5 = 125. The others are not perfect cubes. **Q2.** The cube root of 27 is: (a) 3 (b) 9 (c) 81 (d) 729 **Answer: (a) 3** Explanation: ³√27 = ³√(3³) = 3. **Q3.** If a³ = 216, then a = (a) 6 (b) 8 (c) 10 (d) 12 **Answer: (a) 6** Explanation: 6 × 6 × 6 = 216, so a = 6. **Q4.** Which is NOT a perfect cube? (a) 64 (b) 125 (c) 216 (d) 100 **Answer: (d) 100** Explanation: 64 = 4³, 125 = 5³, 216 = 6³. But 100 ≠ any integer³. **Q5.** ³√(8/27) = (a) 2/3 (b) 8/27 (c) 4/9 (d) 1 **Answer: (a) 2/3** Explanation: ³√(8/27) = ³√8 / ³√27 = 2/3.

Short-answer questions require you to show working and demonstrate understanding of cube root computation and properties. **Q1.** Find ³√512 using prime factorisation. **Solution:** 512 = 2 × 256 = 2 × 2 × 128 = 2 × 2 × 2 × 64 = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 512 = 2⁹ ³√512 = ³√(2⁹) = ³√(2³ × 2³ × 2³) = 2 × 2 × 2 = 8 Alternatively: 512 = 2⁹ = (2³)³, so ³√512 = 2³ = 8. **Q2.** Is 1728 a perfect cube? If yes, find its cube root. **Solution:** Prime factorization of 1728: 1728 ÷ 2 = 864 864 ÷ 2 = 432 432 ÷ 2 = 216 216 ÷ 2 = 108 108 ÷ 2 = 54 54 ÷ 2 = 27 27 ÷ 3 = 9 9 ÷ 3 = 3 3 ÷ 3 = 1 So 1728 = 2⁶ × 3³ = (2²)³ × 3³ = (4 × 3)³ = 12³ Yes, 1728 is a perfect cube, and ³√1728 = 12. **Q3.** Find ³√(-343). **Solution:** 343 = 7³ (since 7 × 7 × 7 = 343) For negative numbers: ³√(-a³) = -³√(a³) = -a ³√(-343) = ³√((-7)³) = -7 **Q4.** Simplify ³√(64 × 125). **Solution:** ³√(64 × 125) = ³√64 × ³√125 = ³√(4³) × ³√(5³) = 4 × 5 = 20 Alternatively: 64 × 125 = 8000 = 20³, so ³√8000 = 20. **Q5.** If a³ = 1000, find the value of a and then calculate 3a + 2. **Solution:** 1000 = 10³, so a = 10 3a + 2 = 3(10) + 2 = 30 + 2 = 32

These questions assess deeper understanding, requiring multi-step reasoning and application of cube properties. **Q1.** Show that 2³ × 3³ is a perfect cube. Find its cube root. **Solution:** 2³ × 3³ = (2 × 3)³ = 6³ = 216 Since 2³ × 3³ = (2 × 3)³, it is a perfect cube. The cube root is: ³√(2³ × 3³) = 2 × 3 = 6 Verification: 6³ = 216, and 2³ × 3³ = 8 × 27 = 216 ✓ **Q2.** The volume of a cubic water tank is 1,000,000 cm³. Find the length of one side of the tank. **Solution:** For a cube, Volume = side³ 1,000,000 = side³ side = ³√1,000,000 Prime factorization: 1,000,000 = 10⁶ = (10²)³ = 100³ side = ³√(100³) = 100 cm Alternatively: 10⁶ = (10²)³, so ³√(10⁶) = 10² = 100 cm. **Q3.** Find ³√(64/729) and express as a fraction in simplest form. **Solution:** ³√(64/729) = ³√64 / ³√729 64 = 4³ and 729 = 9³ ³√64 = 4 and ³√729 = 9 Therefore: ³√(64/729) = 4/9 Verification: (4/9)³ = 64/729 ✓ **Q4.** Which of the following numbers is a perfect cube? 216, 243, 260. Show your reasoning using prime factorisation. **Solution:** 216 = 2³ × 3³ = (2 × 3)³ = 6³ ✓ Perfect cube 243 = 3⁵ = 3³ × 3². Not a perfect cube (exponent of 3 is not divisible by 3) 260 = 4 × 65 = 4 × 5 × 13 = 2² × 5 × 13. Not a perfect cube (exponents are not all divisible by 3) Answer: Only 216 is a perfect cube.

Long-answer questions test integrated understanding, combining cube properties, prime factorisation, and problem-solving. **Q1.** A merchant has 24,389 cubic metres of storage space. He wants to arrange this into a cubic container. Find the dimension of the cubic container. Also, if the cost of construction is ₹50 per metre², calculate the total cost to build all six faces of the cube. **Solution:** Step 1: Find the side length of the cubic container. Volume = 24,389 m³ side³ = 24,389 side = ³√24,389 Prime factorization of 24,389: 24,389 ÷ 29 = 841 841 ÷ 29 = 29 29 ÷ 29 = 1 So 24,389 = 29³ side = ³√(29³) = 29 m Step 2: Calculate surface area and total cost. Surface area of a cube = 6 × side² = 6 × 29² = 6 × 841 = 5,046 m² Total cost = 5,046 × ₹50 = ₹2,52,300 **Q2.** Prove that if m = a³ and n = b³, then mn is also a perfect cube. Take a = 2 and b = 3 to verify your proof. **Solution:** Proof: m = a³ and n = b³ mn = a³ × b³ = (a × b)³ Since mn = (ab)³, it is a perfect cube. Verification with a = 2, b = 3: m = 2³ = 8 n = 3³ = 27 mn = 8 × 27 = 216 (ab)³ = (2 × 3)³ = 6³ = 216 Therefore mn = 216, which is a perfect cube (= 6³) ✓ **Q3.** Find the cube root of 0.000027 and express it in decimal form. **Solution:** 0.000027 = 27 / 1,000,000 = 27 / 10⁶ ³√0.000027 = ³√(27/10⁶) = ³√27 / ³√(10⁶) = ³√(3³) / ³√(10⁶) = 3 / (10²) = 3 / 100 = 0.03 Verification: (0.03)³ = 0.000027 ✓

Higher-order thinking skills questions embed cubes and cube roots in realistic contexts, requiring analysis and synthesis. **Case Study:** A toy company manufactures cubic dice with volumes of 64 cm³, 216 cm³, and 512 cm³. A school wants to order dice such that their edges form an arithmetic sequence. Find: (a) the edges of each die; (b) the common difference; (c) the total volume of all three dice; (d) if a box can hold 30 dice, how many boxes are needed for 1,800 dice? **Solution:** **Step 1: Find the edge of each die using cube roots.** Die 1: Volume = 64 cm³ → edge = ³√64 = ³√(4³) = 4 cm Die 2: Volume = 216 cm³ → edge = ³√216 = ³√(6³) = 6 cm Die 3: Volume = 512 cm³ → edge = ³√512 = ³√(8³) = 8 cm **Step 2: Check if edges form an arithmetic sequence.** Edges: 4, 6, 8 Difference: 6 − 4 = 2, and 8 − 6 = 2 Yes, they form an arithmetic sequence with common difference d = 2 cm. **Step 3: Calculate total volume.** Total volume = 64 + 216 + 512 = 792 cm³ **Step 4: Determine number of boxes needed.** Total dice to order = 1,800 Dice per box = 30 Number of boxes = 1,800 ÷ 30 = 60 boxes **Answers:** (a) Edges are 4 cm, 6 cm, and 8 cm (b) Common difference = 2 cm (c) Total volume = 792 cm³ (d) 60 boxes are needed for 1,800 dice

Solving questions once is not enough—research on learning retention shows that spaced repetition with varied question types solidifies conceptual mastery. At cbsetutor.ai, our AI tutor generates personalized daily drills on Cubes and Cube Roots by analyzing your performance on previous attempts. If you struggle with prime factorisation, the system prioritizes more cube-root-by-factorisation questions. If you excel at perfect cube identification, it escalates to complex reasoning questions (like the case-study above). Each drill session adapts in real-time: if you make an arithmetic error, the AI replays the concept visually before presenting a similar question. Our system also cross-links related topics—when you practice cube roots, the AI reinforces integer exponent rules and number properties, ensuring no knowledge gaps. Parents and teachers appreciate that cbsetutor.ai tracks progress transparently, flagging weak areas with diagnostic reports. Most students see a 1–2 grade improvement in mathematics within 6–8 weeks of consistent use. Start a 3-day free trial at cbsetutor.ai to experience personalized learning—no credit card required, full access to all Class 9 Mathematics chapters.