Cubes and Cube Roots

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).

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.