MP Board Class 8 Mathematics

Chapter: Cube and Cube Roots (घन और घनमूल)

Quick Revision Notes & Formula Sheet

1. Key Definitions (मूल अवधारणाएं)

• Cube of a Number (संख्या का घन): When a number is multiplied by itself three times, the resulting product is called the cube of that number.

  • Formula: Cube of a = a^3 = a × a × a
  • Example: 5^3 = 5 × 5 × 5 = 125

• Perfect Cube (पूर्ण घन): A natural number is called a perfect cube if it is the cube of some natural number.

  • Example: 1, 8, 27, 64, 125, ... are perfect cubes.

• Cube Root (घनमूल): The cube root of a number x is the number whose cube is x. It is denoted by the symbol ∛x or x^(1/3).

  • If: a^3 = b
  • Then: ∛b = a
  • Example: Since 4^3 = 64, ∛64 = 4.

2. Properties of Cubes (घन के गुण)

1. Even & Odd Numbers:

   • Cube of an even number is always even. (e.g., 2^3 = 8, 4^3 = 64)
   • Cube of an odd number is always odd. (e.g., 3^3 = 27, 5^3 = 125)

2. Negative Numbers:

   • Cube of a negative number is always negative.
   • Formula: (-a)^3 = -(a^3)
   • Example: (-3)^3 = -27

3. Units Digit Pattern of Cubes:

Memory Trick: Numbers ending in 0, 1, 4, 5, 6, 9 keep the same unit digit in their cube. For 2 ↔ 8 and 3 ↔ 7, the sum of the digits is 10.

3. Interesting Patterns (रोचक पैटर्न)

• Adding Consecutive Odd Numbers:
  • 1 = 1 = 1^3
  • 3 + 5 = 8 = 2^3
  • 7 + 9 + 11 = 27 = 3^3
  • 13 + 15 + 17 + 19 = 64 = 4^3
  • 21 + 23 + 25 + 27 + 29 = 125 = 5^3

4. Methods to Find Cube Root (घनमूल ज्ञात करने की विधियाँ)

Method 1: Prime Factorization Method (अभाज्य गुणनखंड विधि)

• Step 1: Express the given number as a product of prime factors.

• Step 2: Group the identical prime factors into triplets (groups of 3).

• Step 3: Take one factor from each triplet and multiply them together.

• Example: Find ∛216

  • 216 = (2 × 2 × 2) × (3 × 3 × 3)
  • ∛216 = 2 × 3 = 6

Method 2: Estimation Method (अनुमान विधि)

• Step 1: Form groups of 3 digits starting from the rightmost digit (Unit's place).

• Step 2: The first group gives the unit digit of the cube root.

• Step 3: Find the largest cube less than or equal to the second group to get the tens digit.

• Example: Find ∛857375

  • Group 1: 375 → Ends in 5, so unit digit is 5.
  • Group 2: 857 → Lies between 9^3 = 729 and 10^3 = 1000. Smallest is 9, so tens digit is 9.
  • Answer: ∛857375 = 95

5. Table of Cubes (1 to 15)

6. Important Exam Tips & Problem Types

1. Smallest number to Multiply/Divide:

   • To make a number a perfect cube: Do prime factorization. Find which factor is not in a triplet.
   • To Multiply: Add the missing factors needed to complete the triplet.
   • To Divide: Divide by the extra factors that do not form a triplet.

2. Quick Cube Root Formula for Fractions:

   • ∛(a / b) = ∛a / ∛b

3. Quick Cube Root Formula for Product:

   • ∛(a × b) = ∛a × ∛b