Chapter 6 — Cubes and Cube Roots

Chapter 6 of the Class 8 Maths NCERT textbook, 'Cubes and Cube Roots', teaches students about perfect cubes (numbers obtained when a number is multiplied by itself three times), methods to identify and work with cube numbers using prime factorisation, and finding cube roots through prime factorisation and other techniques.

• What makes a perfect cube — A perfect cube comes from multiplying a number by itself three times, and its prime factorisation always groups every factor in threes — the defining test used to identify cubes and build them.
• Patterns and cube roots — Consecutive odd numbers sum to perfect cubes, units digits follow patterns, and cube roots are found by grouping prime factors into triplets — the inverse of cubing.
• A famous number and applications — The Hardy-Ramanujan number 1729, the smallest expressible as a sum of two cubes in two ways, frames the chapter, alongside practical tasks like counting cuboids needed to form a perfect cube.

1. 01Perfect cubes are numbers obtained when a number is multiplied by itself three times (e.g., 2³ = 8, 3³ = 27)
2. 02In the prime factorisation of a perfect cube, every prime factor appears exactly three times
3. 03The sum of consecutive odd numbers equals perfect cubes (e.g., 1 = 1³; 3 + 5 = 2³; 7 + 9 + 11 = 3³)
4. 04The cube root of a number is the inverse operation of finding its cube (e.g., ∛8 = 2 because 2³ = 8)
5. 05Cube roots can be found by prime factorisation by grouping prime factors into triplets
6. 06The Hardy-Ramanujan Number 1729 is the smallest number expressible as the sum of two cubes in two different ways: 1729 = 12³ + 1³ = 10³ + 9³