Cubes & Cube Roots — Class 8

Compact notes covering cubes, cube roots, methods to find cube roots (prime factorization & estimation), shortcuts, solved examples and practice — mobile-friendly & SEO-optimized.

1. Definitions

Cube: The cube of a number a is a³ = a × a × a. Example: 4³ = 64.

Cube root: The cube root of a number n is a number r such that r³ = n. We write r = ³√n. Example: ³√27 = 3. Cube roots can be positive or negative (since (-3)³ = -27).

2. Perfect Cubes (first 20)

1 = 1³
8 = 2³
27 = 3³
64 = 4³
125 = 5³
216 = 6³
343 = 7³
512 = 8³
729 = 9³
1000 = 10³
1331 = 11³
1728 = 12³
2197 = 13³
2744 = 14³
3375 = 15³
4096 = 16³
4913 = 17³
5832 = 18³
6859 = 19³
8000 = 20³

3. Methods to Find Cube Roots

Prime Factorization Method

Factor the number into primes. If all prime powers are multiples of 3, the number is a perfect cube. Example: 216 = 2³ × 3³ → ³√216 = 2 × 3 = 6.

Estimation Using Nearest Perfect Cubes

Find two consecutive perfect cubes between which the number lies. Example: ³√50 lies between ³√27(=3) and ³√64(=4) → approx 3.68.

Using Cube Root Algorithm (digit grouping)

Similar to the long division method for square roots, cube root extraction uses digit-grouping in threes from the decimal point and a step-wise algorithm to get each digit of the cube root. Practice helps build speed.

4. Shortcuts & Tricks

  • Last digit rule: The last digit of a cube depends only on the last digit of its root. E.g., if a number ends with 7, ³√ ends with 3 because 3³ = 27.
  • To quickly check if a number is a perfect cube, prime-factorize and check exponents mod 3.
  • Use the table of small cubes for quick mental checks (1³ to 20³ memorized for competitive speed).

5. Solved Examples

Example 1: Find ³√343.
343 = 7³ → ³√343 = 7.
Example 2: Is 2197 a perfect cube? Find its cube root.
2197 = 13³ → ³√2197 = 13.
Example 3 (Negative cube root): Find cube root of −125.
³√(−125) = −5 because (−5)³ = −125.
Example 4 (Estimation): Estimate ³√150.
Nearest cubes: 125(=5³) and 216(=6³). So ³√150 ≈ 5.3 (approx). Use interpolation for better estimate.

6. Practice Questions (with answers)

  1. Find cube root of 512.
    Answer
    ³√512 = 8
  2. Is 1000 a perfect cube?
    Answer
    Yes, 1000 = 10³ → ³√1000 = 10
  3. Find cube root of −27.
    Answer
    ³√(−27) = −3
  4. Which number between 20 and 30 is a perfect cube?
    Answer
    27 = 3³

Practice prime factorization for larger numbers and use the cubes table for quick recognition in exams.