Introduction - Cubes and Cube Roots - Chapter 6 - NCERT Class 8th Maths

Cubes and Cube Roots: Properties, Patterns, and Applications

The cube of a number is the product obtained when the number is multiplied by itself three times. If n is a number, its cube is n³ = n × n × n. For example, the cube of 2 is 2³ = 2×2×2 = 8, and the cube of 5 is 5³ = 125. The cube root is the inverse operation — finding the number whose cube equals a given number. The cube root of 8 is 2 (written as ∛8 = 2), and the cube root of 125 is 5. This chapter in CBSE Class 8 Mathematics explores the patterns and properties of cubes, methods for finding cube roots through prime factorisation and estimation, and the applications of these concepts in mensuration and real-world problems.

Every number has a unique cube. Cubes of natural numbers exhibit interesting patterns. The cubes of odd numbers are always odd, and cubes of even numbers are always even. The sum of the cubes of the first n natural numbers equals the square of the sum of the first n natural numbers: 1³ + 2³ + 3³ + ... + n³ = (1+2+3+...+n)² = [n(n+1)/2]². For example, 1³ + 2³ + 3³ = 1 + 8 + 27 = 36 = 6² = (1+2+3)². Cubes of numbers ending in a specific digit always end in a predictable digit: if a number ends in 0, its cube ends in 0; if it ends in 1, its cube ends in 1; 2→8, 3→7, 4→4, 5→5, 6→6, 7→3, 8→2, 9→9. This property is useful for estimating cube roots. For example, the cube of 13 is 2197. Since 13 ends in 3, its cube ends in 7 (2197 ends in 7 ✓). The unit digit pattern is symmetric: (0→0, 1→1), (2→8, 8→2), (3→7, 7→3), (4→4, 6→6), (5→5, 9→9).

Finding the cube root of a perfect cube by prime factorisation: express the number as a product of its prime factors and then group the factors into triples of identical prime numbers. The cube root is the product of one factor from each group. For example, to find ∛1728: 1728 = 2×2×2 × 2×2×2 × 3×3×3 = 2³×2³×3³, so ∛1728 = 2×2×3 = 12. For 512: 512 = 2³×2³×2³ = 8×8×8, so ∛512 = 8. If a number cannot be expressed as a product of prime factor triples, it is not a perfect cube. For estimating cube roots of larger numbers, the unit digit of the cube gives the unit digit of the cube root (using the pattern above), and the number formed by the remaining digits (after removing the last three digits) helps determine the tens digit by finding the largest cube less than or equal to that number. For example, to find ∛68921: last digit 1 → cube root ends in 1. Remove last three digits: 68. The largest cube ≤ 68 is 64 = 4³, so the cube root is 41 (since 41³ = 68921). For negative numbers, the cube root is negative: ∛(−125) = −5 because (−5)³ = −125. Cube and cube roots are used in volume calculations for cubes and cuboids, in scaling problems where linear dimensions are scaled by a factor k and the volume scales by k³, and in scientific contexts where quantities vary as the cube of a variable — for example, the power required to overcome air resistance at high speed varies as the cube of the velocity.

• Cube of a number n: n³ = n×n×n; cube root is the inverse: if a³ = b, then ∛b = a.
• Unit digit pattern of cubes: 0→0, 1→1, 2→8, 3→7, 4→4, 5→5, 6→6, 7→3, 8→2, 9→9 — useful for estimation.
• Prime factorisation method: group prime factors into triples; multiply one factor from each triple to get the cube root.
• Estimation method for large cubes: use the last digit to find the unit digit, and remaining digits to find the tens digit by comparing with cubes of tens numbers.
• Cube roots of negative numbers are negative: ∛(−a) = −∛a; used in volume calculations and scaling problems where volume ∝ (linear dimension)³.