Class 8 Notes

Class 8 Mathematics Notes Chapter 7 (Cubes and Cube Roots) – Mathematics Book

Philoid

Philoid

— 7 min read

Mathematics
Alright class, let's focus on Chapter 7: Cubes and Cube Roots. This is an important topic, not just for your class exams but also for various government competitive exams where basic numerical ability is tested. Pay close attention to the concepts and methods.

Chapter 7: Cubes and Cube Roots - Detailed Notes for Competitive Exams

1. What are Cubes?

  • When a number is multiplied by itself three times, the resulting number is called its cube.
  • If 'x' is a number, its cube is x × x × x = x³.
  • Examples:
    • Cube of 2 = 2³ = 2 × 2 × 2 = 8
    • Cube of 5 = 5³ = 5 × 5 × 5 = 125
    • Cube of (-3) = (-3)³ = (-3) × (-3) × (-3) = -27

2. Perfect Cubes (or Cube Numbers)

  • A natural number 'n' is a perfect cube if it is the cube of some natural number 'm'. That is, n = m³.
  • Examples: 1 (1³), 8 (2³), 27 (3³), 64 (4³), 125 (5³), 216 (6³), 343 (7³), 512 (8³), 729 (9³), 1000 (10³) are the first 10 perfect cubes.
  • Key for Exams: You should try to memorize cubes up to at least 15 (or even 20) for faster calculations.
    • 11³ = 1331
    • 12³ = 1728
    • 13³ = 2197
    • 14³ = 2744
    • 15³ = 3375

3. Properties of Cube Numbers

  • (i) Cubes of Even/Odd Numbers:

    • The cube of an even number is always even. (e.g., 4³ = 64, 6³ = 216)
    • The cube of an odd number is always odd. (e.g., 3³ = 27, 7³ = 343)

  • (ii) Unit Digits of Cubes: The unit digit of the cube of a number depends on the unit digit of the original number. This is very useful for quick checks and estimations.

    Unit digit of number Unit digit of its cube Example
    0 0 10³ = 1000
    1 1 11³ = 1331
    2 8 12³ = 1728
    3 7 13³ = 2197
    4 4 14³ = 2744
    5 5 15³ = 3375
    6 6 16³ = 4096
    7 3 17³ = 4913
    8 2 18³ = 5832
    9 9 19³ = 6859
    • Important Observation: Numbers ending with digits 2, 3, 7, or 8 are never perfect cubes. A number ending in 0, 1, 4, 5, 6, or 9 might be a perfect cube.
    • Zeros: A perfect cube always ends with a number of zeros that is a multiple of 3 (e.g., 1000 (3 zeros), 8000000 (6 zeros)). 100 or 10000 are not perfect cubes.

  • (iii) Sum of Consecutive Odd Numbers: Perfect cubes can be expressed as the sum of consecutive odd numbers.

    • 1³ = 1
    • 2³ = 8 = 3 + 5
    • 3³ = 27 = 7 + 9 + 11
    • 4³ = 64 = 13 + 15 + 17 + 19
    • (This is an interesting property, less commonly tested directly in MCQs but good to know).

4. Prime Factorisation Method to Check for Perfect Cubes

  • This is a fundamental method.
  • Steps:
    1. Find the prime factors of the given number.
    2. Group the identical prime factors in triplets (groups of three).
    3. Check: If all prime factors can be grouped into triplets with no factor left over, the number is a perfect cube. Otherwise, it is not.
  • Example 1: Is 216 a perfect cube?
    • Prime Factorisation: 216 = 2 × 108 = 2 × 2 × 54 = 2 × 2 × 2 × 27 = 2 × 2 × 2 × 3 × 3 × 3
    • Grouping: 216 = (2 × 2 × 2) × (3 × 3 × 3) = 2³ × 3³
    • Conclusion: All factors form triplets. So, 216 is a perfect cube. (It's 6³)
  • Example 2: Is 500 a perfect cube?
    • Prime Factorisation: 500 = 2 × 250 = 2 × 2 × 125 = 2 × 2 × 5 × 5 × 5
    • Grouping: 500 = (2 × 2) × (5 × 5 × 5)
    • Conclusion: The factor 2 does not form a triplet (only two 2s). So, 500 is not a perfect cube.

5. Finding the Smallest Number to Multiply/Divide to Get a Perfect Cube

  • This is a common exam question type. Use the prime factorisation method.
  • To Multiply:
    1. Find the prime factorisation.
    2. Identify the factors that do not form a triplet.
    3. Determine the additional factors needed to complete the triplet(s).
    4. The product of these additional factors is the smallest number to multiply.
    • Example: Smallest number to multiply 500 (2 × 2 × 5 × 5 × 5) to make it a perfect cube?
      • We need one more '2' to complete the triplet of 2s.
      • Smallest number = 2. (500 × 2 = 1000 = 10³)
  • To Divide:
    1. Find the prime factorisation.
    2. Identify the factors that are "left over" after forming triplets.
    3. The product of these leftover factors is the smallest number to divide.
    • Example: Smallest number to divide 8640 to make it a perfect cube?
      • 8640 = 10 × 864 = 2 × 5 × 8 × 108 = 2 × 5 × 2³ × 2² × 3³ = 2⁶ × 3³ × 5¹
      • Grouping: (2×2×2) × (2×2×2) × (3×3×3) × 5
      • The factor '5' is left over.
      • Smallest number = 5. (8640 / 5 = 1728 = 12³)

6. Cube Roots

  • Finding the cube root is the inverse operation of finding the cube.
  • The symbol for cube root is ³√ .
  • If m³ = n, then ³√n = m.
  • Examples:
    • ³√8 = 2 (since 2³ = 8)
    • ³√125 = 5 (since 5³ = 125)
    • ³√(-27) = -3 (since (-3)³ = -27)

7. Methods to Find Cube Roots

  • (i) Prime Factorisation Method: (Works for any perfect cube)
    1. Find the prime factors of the number.
    2. Group the factors in triplets. (The number must be a perfect cube for this method to give a whole number answer).
    3. Take one factor from each triplet.
    4. Multiply these chosen factors together to get the cube root.
    • Example: Find ³√1728
      • 1728 = 2 × 864 = 2 × 2 × 432 = 2 × 2 × 2 × 216 = 2³ × 6³ = 2³ × (2×3)³ = 2³ × 2³ × 3³
      • Grouping: (2 × 2 × 2) × (2 × 2 × 2) × (3 × 3 × 3)
      • Take one factor from each triplet: 2, 2, 3
      • Cube root = 2 × 2 × 3 = 12.
  • (ii) Estimation Method: (Useful for finding cube roots of perfect cubes quickly, especially larger ones)
    1. Step 1: Group the digits of the number in threes, starting from the right (unit's place). The leftmost group can have 1, 2, or 3 digits.
      • Example: Find ³√17576. Grouping: 17 576
    2. Step 2: Look at the rightmost group (e.g., 576). The unit digit of this group determines the unit digit of the cube root.
      • 576 ends in 6. Which number's cube ends in 6? Only 6³ = 216 ends in 6.
      • So, the unit digit of the cube root is 6.
    3. Step 3: Look at the next group (leftmost group) (e.g., 17). Find the largest number whose cube is less than or equal to this group.
      • We need a number 'a' such that a³ ≤ 17.
      • 1³ = 1, 2³ = 8, 3³ = 27.
      • The largest cube less than or equal to 17 is 8, which is 2³.
      • So, the tens digit of the cube root is 2.
    4. Step 4: Combine the digits found.
      • The cube root is 26. (Check: 26³ = 17576)
    • Example 2: Find ³√117649
      • Grouping: 117 649
      • Right group 649 ends in 9. Unit digit of cube root is 9 (since 9³ = 729).
      • Left group 117. We need a³ ≤ 117. 4³ = 64, 5³ = 125. So, we take 4.
      • Cube root is 49.

Summary for Quick Revision:

  • Cube: x³ = x × x × x
  • Perfect Cube: A number that is the cube of an integer. Check using prime factorisation (triplets).
  • Properties: Even³=Even, Odd³=Odd. Know unit digits (2↔8, 3↔7, others same). Ending zeros must be multiple of 3.
  • Cube Root: ³√n. Inverse of cube.
  • Finding Cube Root: Prime Factorisation (universal for perfect cubes) or Estimation (quick for perfect cubes).
  • Exam Questions: Check if perfect cube, find smallest multiplier/divisor, find cube root.

Now, let's test your understanding with some multiple-choice questions.

Multiple Choice Questions (MCQs)

  1. Which of the following numbers is a perfect cube?
    (a) 10000
    (b) 3375
    (c) 1225
    (d) 400

  2. The cube of an odd natural number is always:
    (a) Even
    (b) Odd
    (c) May be even or odd
    (d) Prime

  3. What is the unit digit of the cube of 77?
    (a) 7
    (b) 9
    (c) 3
    (d) 1

  4. What is the value of ³√512?
    (a) 6
    (b) 7
    (c) 8
    (d) 9

  5. By what smallest natural number should 392 be multiplied so that the product is a perfect cube?
    (a) 2
    (b) 3
    (c) 7
    (d) 49

  6. Find the cube root of 13824 using prime factorisation.
    (a) 22
    (b) 24
    (c) 26
    (d) 28

  7. Which of the following numbers cannot be a perfect cube?
    (a) 6859
    (b) 9261
    (c) 4096
    (d) 7987

  8. The value of ³√(-2197) is:
    (a) 13
    (b) -13
    (c) 17
    (d) -17

  9. By what smallest natural number should 1080 be divided so that the quotient is a perfect cube?
    (a) 5
    (b) 10
    (c) 4
    (d) 40

  10. Estimate the cube root of 91125.
    (a) 35
    (b) 45
    (c) 55
    (d) 65

Answer Key for MCQs:

  1. (b) 3375 (15³)
  2. (b) Odd
  3. (c) 3 (Unit digit of 77 is 7, 7³ = 343, unit digit is 3)
  4. (c) 8 (8 × 8 × 8 = 512)
  5. (c) 7 (392 = 2 × 2 × 2 × 7 × 7 = 2³ × 7². Need one more 7)
  6. (b) 24 (13824 = 2⁹ × 3³ = (2³×2³×2³) × 3³ = (8×8×8) × 27. Cube root = 2×2×2×3 = 24)
  7. (d) 7987 (Perfect cubes cannot end in 7)
  8. (b) -13 ( (-13) × (-13) × (-13) = -2197)
  9. (d) 40 (1080 = 108 × 10 = 2² × 3³ × 2 × 5 = 2³ × 3³ × 5. Leftover factors are 5. Wait, 1080 = 27 * 40 = 3^3 * 2^3 * 5. Leftover factor is 5. Let's recheck prime factorization: 1080 = 10 * 108 = 2 * 5 * 4 * 27 = 2 * 5 * 2^2 * 3^3 = 2^3 * 3^3 * 5. The leftover factor is 5. Smallest number to divide is 5. Let me re-evaluate the options and my calculation. 1080 / 5 = 216 = 6³. So 5 is the answer. Option (a) is 5. Let me check option (d) 40. 1080 / 40 = 108 / 4 = 27 = 3³. So 40 is also a number you can divide by. The question asks for the smallest natural number. 5 is smaller than 40. Hmm, maybe my prime factorization was wrong. 1080 = 10 * 108 = 2 * 5 * 2 * 54 = 2 * 5 * 2 * 2 * 27 = 2 * 5 * 2 * 2 * 3 * 3 * 3 = 2³ * 3³ * 5. Yes, the only leftover factor is 5. The smallest number to divide is 5. The provided option (d) 40 must be based on a different interpretation or maybe an error in the question/options often seen in exam prep material. Let's assume the question meant "by what number from the options...". If we divide by 40, we get 27 (3³). If we divide by 5, we get 216 (6³). Both result in perfect cubes. 5 is the smallest natural number. 40 is also a natural number. The question asks for the smallest natural number. That should be 5. Let's stick to the method. The leftover prime factor is 5. So the answer should be 5. I will mark (a) as the correct answer based on the method. Self-correction: Re-read the question. Smallest natural number. Prime factorization is 2³ * 3³ * 5. The factor that is not in a triplet is 5. So, divide by 5. 1080/5 = 216 = 6³. The smallest number is 5. Answer is (a).
  10. (b) 45 (Grouping: 91 125. Unit digit is 5. For 91, 4³=64, 5³=125. Take 4. So, 45.)

(Self-correction on Q9: The method clearly points to 5. The smallest number composed of leftover factors after forming triplets is 5. 1080 = (2x2x2) x (3x3x3) x 5. Leftover is 5. So divide by 5. The answer is (a)).

Make sure you practice these concepts thoroughly, especially the prime factorisation method and the estimation technique for cube roots. Good luck with your preparation!

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