Cubes and Cube Roots — Class 8 Mathematics
1. Perfect Cubes
A PERFECT CUBE is a number that is the CUBE of an integer. n³ = n × n × n. Examples: 1³=1, 2³=8, 3³=27, 4³=64, 5³=125, 6³=216, 7³=343, 8³=512, 9³=729, 10³=1000. First 10 cubes should be memorised.
The Hardy-Ramanujan Number — 1729
'1729 is the SMALLEST number expressible as the SUM of TWO CUBES in TWO DIFFERENT WAYS.' 1729 = 1³ + 12³ = 1 + 1728. 1729 = 9³ + 10³ = 729 + 1000. Story: G.H. Hardy visited Ramanujan in a taxi numbered 1729. Hardy remarked it was a 'dull number.' Ramanujan immediately replied: 'No, Hardy! It is a very interesting number...' This story is part of Indian mathematical FOLKLORE.
Properties of Perfect Cubes
- If a number ends in 1, its cube ends in 1. Ends in 2 → cube ends in 8. 3 → 7. 4 → 4. 5 → 5. 6 → 6. 7 → 3. 8 → 2. 9 → 9. 0 → 0.
- Cubes of EVEN numbers are EVEN. Cubes of ODD numbers are ODD.
- Cubes of negative numbers are NEGATIVE: (−2)³ = −8, (−3)³ = −27.
- A perfect cube has each prime factor appearing in GROUPS of THREE in prime factorisation.
Patterns with Cubes
- Sum of cubes of first n natural numbers = [n(n+1)/2]². 1³+2³+3³ = 1+8+27 = 36 = 6² = (1+2+3)².
- Difference of cubes: (n+1)³ − n³ = 3n² + 3n + 1.
2. Finding Cube Roots
The CUBE ROOT of n is the number m such that m³ = n. Symbol: ³√n. Unlike square roots, cube roots of NEGATIVE numbers ARE real: ³√(−8) = −2 because (−2)³ = −8.
Method — Prime Factorisation
Express number as product of primes. Make GROUPS of THREE identical factors. Take ONE factor from each group. Multiply.
Example — ³√3375: 3375 = 3×1125 = 3×3×375 = 3²×3×125 = 3³×5³. Groups: (3×3×3)×(5×5×5). Take: 3×5 = 15. ³√3375 = 15.
Example — ³√(−1728) : First ³√1728: 1728 = 2×864 = 2×2×432 = 2³×216 = 2³×6³ = 12³. ³√1728 = 12. So ³√(−1728) = −12.
Example — ³√13824: 13824 = 2⁹×3³ = (2³)³×3³. Take: 2³×3 = 8×3 = 24. ³√13824 = 24.
3. Estimating Cube Roots
For numbers that are NOT perfect cubes, estimate using the nearest perfect cubes. Method: Group digits in THREES from the right. For the first group from the right, its cube root gives the unit digit. For the leftmost group, find the largest cube ≤ it.
Example — ³√17576: Groups: 17 and 576. Unit digit of 576: ends in 6 → cube root ends in 6 (since 6³=216 ends in 6). Leftmost: 17. Cubes: 1³=1, 2³=8, 3³=27 (too big). So tens digit = 2. Cube root ≈ 26. Check: 26³ = 17576 ✓.
4. Comparing Properties — Square vs Cube
| Property | Squares | Cubes |
|---|---|---|
| Sign | Always non-negative | Same sign as base |
| n=2 | 4 | 8 |
| n=−2 | 4 | −8 |
| Root symbol | √ | ³√ |
| Method | Pair factors | Group in 3 |
'The key difference: (−2)² = +4, but (−2)³ = −8. Cubes preserve SIGN — this is very useful in algebra.'
5. Common Mistakes
- Writing ³√(−8) as imaginary/non-existent: Cube roots of NEGATIVE numbers ARE real. ³√(−8) = −2.
- Confusing (a+b)³ with a³+b³: (2+3)³ = 5³ = 125. 2³+3³ = 8+27 = 35. They are DIFFERENT.
- In prime factorisation, making pairs instead of triplets: Cube roots need groups of THREE, not two.
6. AP Exam Focus
| Topic | Marks |
|---|---|
| Prime factorisation for cube root | 3-4 |
| Cube root of negative integers | 1-2 |
| Properties of cubes | 2-3 |
| Word problems involving volumes | 3-4 |
Key Exam Tips
- For cube root by prime factorisation: show the factor tree clearly. Circle groups of three. Write '³√____ = ____' in the final step.
- Remember: a cube has side = ³√Volume. Surface area = 6s². Many word problems combine these.
- The Hardy-Ramanujan number (1729) is a favourite question for 1-2 marks. Know both representations.
Quick Self-Test
- Cube of −5? (Answer: −125.)
- ³√5832 = ? (Answer: 5832 = 2³×3⁶ = 18³ → 18.)
- Sum of cubes of first 3 natural numbers? (Answer: 1+8+27 = 36 = 6².)
- Is 8000 a perfect cube? (Answer: 8000 = 8×1000 = 2³×10³ = 20³. Yes.)
- ³√(−343) = ? (Answer: −7, since (−7)³ = −343.)
More Worked Examples
Example — Cube Root by Estimation: Find ³√12167. Group from right in threes: 12 and 167. Unit digit of 167 is 7 → cube root ends in 3 (because 3³=27 ends in 7). Leftmost group: 12. Largest cube ≤ 12: 2³=8. So tens = 2. Cube root ≈ 23. Check: 23³ = 12167 ✓.
Example — Volume Application: A cubical box has volume 1728 cm³. Find its side and TSA. Side = ³√1728 = ³√(12³) = 12 cm. TSA = 6a² = 6×144 = 864 cm².
Example — Smallest Number to Make a Perfect Cube: What smallest number must 256 be multiplied by to get a perfect cube? Prime factorisation: 256 = 2⁸. To form groups of 3: 2⁸ = (2³)² × 2². We need one more 2 to make the last group of 3. So multiply by 2: 256×2=512 = 2⁹ = (2³)³ = 8³. Answer: 2.
Example — Successive Cubes Pattern: The difference between cubes of consecutive numbers follows a pattern. 2³−1³ = 7. 3³−2³ = 19. 4³−3³ = 37. 5³−4³ = 61. Differences: 7, 19, 37, 61... Second differences: 12, 18, 24... increase by 6 each time. 'Cubes grow MUCH faster than squares. 10³=1000 vs 10²=100.'
Common Exam Errors
- Writing cube root of a negative number as "not possible" — It IS possible in real numbers. ³√(−64) = −4.
- In prime factorisation for cube roots, making groups of TWO instead of THREE. Cube root = groups of 3 identical factors.
- Forgetting to include the negative sign when taking cube root of a negative: ³√(−27a³) = −3a.
Connection to Class 9 — Surds and Real Numbers
Cube roots of non-perfect cubes are IRRATIONAL numbers. ³√2, ³√3, ³√4 — none can be expressed as p/q. 'Just like √2 is irrational, ³√2 is also irrational. This concept extends to Class 9 real numbers.'
Cube Roots by Estimation — More Practice
Find ³√15625. No grouping needed — notice 25³=15625 (25×25×25=15625). Or group: 15 and 625. 625 ends in 5 → cube root ends in 5. 15: 2³=8, 3³=27 (too big). Tens=2. Cube root=25.
Find ³√21952. Groups: 21 and 952. 952 ends in 2 → cube root ends in 8. 21: 2³=8, 3³=27 (too big). Tens=2. Cube root=28. Check: 28³=21952 ✓.
Find ³√110592. Groups: 110 and 592. 592 ends in 2 → ends in 8. 110: 4³=64, 5³=125 (too big). Tens=4. Cube root=48. Check: 48³=110592 ✓.
Cube Roots in Geometry — Volume Applications
Example: Three cubes of sides 3 cm, 4 cm, and 5 cm are melted and recast into a single cube. Find the side. Volumes: 27+64+125 = 216 cm³. Side = ³√216 = 6 cm. 'When melting and recasting, TOTAL VOLUME is conserved. The new side is the cube root of the total volume.'
Example: A cuboid 8cm×12cm×18cm is melted into a cube. Side of cube? Volume = 8×12×18 = 1728 cm³. Side = ³√1728 = 12 cm. TSA of cube = 6×12² = 864 cm².
Cubes and Cube Roots in Indian Mathematics
Aryabhata (476 CE) gave methods for finding cube roots in his work 'Aryabhatiya.' Indian mathematicians developed algorithms for cube roots centuries before similar methods appeared in Europe. The cubic equation x³=2 (doubling the cube) was one of the three classical problems of Greek geometry. 'Ancient Indian mathematicians like Aryabhata and Brahmagupta made significant contributions to understanding cubes, cube roots, and cubic equations.'
AP Exam Question Pattern
- 'Find the cube root of ___ by prime factorisation' → Show factor tree. Group in 3s. Write result.
- 'Is ___ a perfect cube?' → Factorise. Check if every prime factor appears in groups of 3.
- 'Find the smallest number to multiply/divide to get a perfect cube' → Factorise. Find unpaired factors (appearing 1 or 2 times in groups of 3).
- Volume word problems → Use volume formula. Cube root for side length.
