Class 8 Notes - Cubes and Cube Roots

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1. Introduction to Cubes and Cube Roots

In this chapter, students will learn about cubes, cube roots, and their properties. Understanding cubes and cube roots is essential for solving higher-level algebraic and arithmetic problems.

2. What is a Cube?

  • The cube of a number is that number multiplied by itself three times.
  • If a is a number, then its cube is a × a × a = a³.
  • Examples:
    • 2³ = 2 × 2 × 2 = 8
    • 5³ = 5 × 5 × 5 = 125
    • (-3)³ = -3 × -3 × -3 = -27

3. Properties of Cubes

  • The cube of a positive number is positive.
  • The cube of a negative number is negative.
  • The cube of zero is zero.
  • The cube of an even number is even; the cube of an odd number is odd.

4. Perfect Cubes

  • A number is a perfect cube if it is the cube of a whole number.
  • Examples of perfect cubes: 1, 8, 27, 64, 125, 216, 343, 512, 729, 1000
  • For example, 27 is a perfect cube because 3 × 3 × 3 = 27.

5. Cube Roots

  • The cube root of a number is a value that, when multiplied by itself three times, gives the original number.
  • Cube root of a is written as ∛a.
  • Examples:
    • ∛27 = 3, because 3 × 3 × 3 = 27
    • ∛64 = 4, because 4 × 4 × 4 = 64
    • ∛-125 = -5, because (-5) × (-5) × (-5) = -125

6. Finding Cube Roots

  • By Prime Factorization: Break the number into prime factors and group them in triples.
  • By Estimation: Estimate between which two perfect cubes the number lies.
  • By Using Cube Root Tables (for larger numbers).
Example: Find the cube root of 216.
Solution: 216 = 2 × 2 × 2 × 3 × 3 × 3 = (2 × 3) × (2 × 3) × (2 × 3) = 6 × 6 × 6
So, ∛216 = 6.

7. Applications of Cubes and Cube Roots

  • Finding the volume of cubes and cuboids.
  • Solving algebraic equations involving cubes and cube roots.
  • Used in geometry, physics, and real-life measurements.

8. Common Mistakes

  • Confusing squares and cubes.
  • Forgetting that the cube root of a negative number is negative.
  • Not grouping prime factors correctly when finding cube roots.

9. Practice Questions

  1. Find the cube of 7.
  2. Is 1000 a perfect cube? If yes, what is its cube root?
  3. Find the cube root of 512.
  4. Which number should be multiplied by 54 to make it a perfect cube?
  5. If the volume of a cube is 343 cm³, what is the length of its side?

10. Summary

  • The cube of a number is that number multiplied by itself three times.
  • The cube root of a number is a value which, when cubed, gives the original number.
  • Perfect cubes and cube roots are useful in many mathematical and real-life problems.