Cubes and Cube Roots (NCERT Ch. 7)

• Identify perfect cubes and recognise cube number patterns.
• Find the cube root of a perfect cube using prime factorisation.

1. A perfect cube is a number that is the cube of a whole number (1, 8, 27, 64, 125, ...).
2. Cube root is the inverse of cubing: if n³ = m, then ∛m = n.
3. Prime factorisation method for cube roots: group prime factors in triples; the cube root is the product of one factor from each triple.
4. Sum of consecutive odd numbers can generate cubes in special patterns.

Find the cube root of 216 using prime factorisation.

216 = 2 x 2 x 2 x 3 x 3 x 3 = (2x2x2) x (3x3x3)
Triples: (2,2,2) and (3,3,3) -> take one from each: 2 x 3 = 6
∛216 = 6

• Confusing square roots (pairs of 2) with cube roots (triples of 3).
• Forgetting that for a perfect cube, EVERY prime factor must appear in a multiple of 3.
• Mixing up cubing (n³ = n x n x n) with simple multiplication by 3 (n x 3).

• Is 100 a perfect cube?
• Find ∛125.

Open the Practice tab for graded questions on Cubes and Cube Roots (NCERT Ch. 7).