Cubes and Cube Roots
1. Perfect Cubes
A perfect cube is the cube of an integer.
1³ = 1, 2³ = 8, 3³ = 27, 4³ = 64, 5³ = 125, 6³ = 216, 7³ = 343, 8³ = 512, 9³ = 729, 10³ = 1000, 11³ = 1331, 12³ = 1728, 13³ = 2197, 14³ = 2744, 15³ = 3375, 16³ = 4096, 20³ = 8000, 25³ = 15625, 30³ = 27000.
'Memorise cubes up to 12 — they help in factorisation and volume problems.'
2. Properties of Cubes
| Property | Example |
|---|---|
| Cube of an EVEN number is EVEN | 4³ = 64 (even) |
| Cube of an ODD number is ODD | 5³ = 125 (odd) |
| Cube of a NEGATIVE number is NEGATIVE | (—3)³ = —27 |
| Cube of a number ending in 0 ends in 000 | 10³ = 1000 |
| Cube of a number ending in 1 ends in 1 | 11³ = 1331 |
| Cube of a number ending in 4 ends in 4 | 4³ = 64 |
| Cube of a number ending in 5 ends in 125 | 5³ = 125 |
| Cube of a number ending in 6 ends in 6 | 6³ = 216 |
| Cube of a number ending in 9 ends in 9 | 9³ = 729 |
'The last digit of a cube UNIQUELY determines the last digit of the base. This helps in estimation.'
3. Cube Roots by Prime Factorisation
To find the cube root of a perfect cube:
- Write the number as a product of its prime factors.
- GROUP the factors into TRIPLETS of identical factors.
- Take ONE factor from each triplet.
- MULTIPLY those factors.
Worked Example: Find ∛3375
3375 = 3 × 3 × 3 × 5 × 5 × 5 = 3³ × 5³ Triplets: (3×3×3), (5×5×5) ∛3375 = 3 × 5 = 15
Worked Example: Find ∛10648
10648 = 2 × 2 × 2 × 11 × 11 × 11 = 2³ × 11³ ∛10648 = 2 × 11 = 22
4. Cube Root by Estimation
For numbers that are perfect cubes but large, use estimation:
- Group the digits in THREES from the RIGHT.
- Look at the LAST digit to determine the LAST digit of the cube root.
- Look at the remaining group to determine the FIRST digit.
Worked Example: Find ∛12167
Group: 12 | 167 Last digit ends in 7 → cube root ends in 3. Remaining: 12. 2³ = 8 < 12 < 3³ = 27. So the tens digit is 2. Therefore ∛12167 = 23.
Check: 23³ = 12167 ✓
5. The Hardy-Ramanujan Number — 1729
1729 is the SMALLEST number that can be expressed as the SUM of TWO CUBES in TWO DIFFERENT ways:
1729 = 12³ + 1³ = 12 × 12 × 12 + 1 × 1 × 1 = 1728 + 1 = 1729 1729 = 9³ + 10³ = 9 × 9 × 9 + 10 × 10 × 10 = 729 + 1000 = 1729
'When the mathematician G.H. Hardy visited Srinivasa Ramanujan in the hospital, he said the taxi number 1729 was dull. Ramanujan instantly replied that 1729 is very interesting — it is the smallest number expressible as the sum of two cubes in two different ways.'
6. Cubes of Fractions and Decimals
(a/b)³ = a³/b³
Example: (2/3)³ = 8/27
(0.1)³ = 0.001, (0.2)³ = 0.008, (0.3)³ = 0.027, (0.4)³ = 0.064, (0.5)³ = 0.125, (1.5)³ = 3.375
Common Mistakes and Fixes
| Mistake | Fix |
|---|---|
| '√[3]{a + b} = ∛a + ∛b' | FALSE! ∛(8 + 27) = ∛35 ≠ ∛8 + ∛27 = 2 + 3 = 5 |
| 'Confusing squares and cubes' | Square groups in PAIRS. Cube groups in TRIPLETS. They are DIFFERENT |
| 'Negative numbers cannot have cube roots' | They CAN! ∛(—8) = —2 because (—2)³ = —8 |
| 'Estimating 0 at the end of a cube root' | If a cube ends in 000, the cube root ends in 0 |
ICSE Exam Focus (4–5 marks)
- 2-mark questions: Finding cube roots of small numbers by prime factorisation
- 3-mark questions: Estimation method or checking if a number is a perfect cube
- 4-mark questions: Word problems involving volume of cubes
- 5-mark questions: Combined with exponents — simplify expressions with cube roots
Self-Test
Q1. Is 729 a perfect cube? If yes, find its cube root. A1. 729 = 3 × 3 × 3 × 3 × 3 × 3 = 3⁶ = (3²)³ = 9³. Yes, it is a perfect cube. ∛729 = 9.
Q2. Find the smallest number by which 256 must be MULTIPLIED to get a perfect cube. A2. 256 = 2⁸ = 2³ × 2³ × 2². We need one more '2' to complete a triplet. Multiply by 2: 256 × 2 = 512 = 8³.
Q3. Evaluate: ∛(64/343) A3. ∛64 = 4, ∛343 = 7. So ∛(64/343) = 4/7.
Q4. Find ∛4913 by estimation. A4. Group: 4 | 913. Last digit 3 → cube root ends in 7. Remaining 4: 1³ = 1 < 4 < 2³ = 8 → tens digit = 1. ∛4913 = 17. Check: 17³ = 4913 ✓.
Q5. Show that 1729 can be expressed as sum of two cubes in two different ways. A5. 1729 = 12³ + 1³ = 1728 + 1. Also 1729 = 10³ + 9³ = 1000 + 729.
Q6. The volume of a cube is 13824 cm³. Find its side. A6. ∛13824. 13824 = 2⁹ × 3³ = (2³)³ × 3³ = (8×3)³ = 24³. Side = 24 cm.
