Power and Roots
O Level and IGCSE Mathematics Cheat Sheet – 1.3 Powers and Roots
Squares and Square Roots (1² to 15²)
| Number | Square (n²) | Square Root (√n²) |
|---|---|---|
| 1 | 1 | 1 |
| 2 | 4 | 2 |
| 3 | 9 | 3 |
| 4 | 16 | 4 |
| 5 | 25 | 5 |
| 6 | 36 | 6 |
| 7 | 49 | 7 |
| 8 | 64 | 8 |
| 9 | 81 | 9 |
| 10 | 100 | 10 |
| 11 | 121 | 11 |
| 12 | 144 | 12 |
| 13 | 169 | 13 |
| 14 | 196 | 14 |
| 15 | 225 | 15 |
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Cubes and Cube Roots
| Number | Cube (n³) | Cube Root (∛n³) |
|---|---|---|
| 1 | 1 | 1 |
| 2 | 8 | 2 |
| 3 | 27 | 3 |
| 4 | 64 | 4 |
| 5 | 125 | 5 |
| 10 | 1000 | 10 |
Other Powers
- n⁴ means n × n × n × n
Example: 2⁴ = 16 - n⁵ means n multiplied by itself 5 times
Example: 3⁵ = 243
Other Roots
- ⁿ√x means the number that, when raised to power n, equals x.
Example: ⁴√81 = 3 because 3⁴ = 81 - Fractional powers:
- n^(1/2) = √n
- n^(1/3) = ∛n
- n^(m/n) = (ⁿ√n)ᵐ
Written and Compiled By Sir Hunain Zia, World Record Holder With 154 Total A Grades, 7 Distinctions and 11 World Records For Educate A Change O Level And IGCSE Mathematics Full Scale Course
Examples
- √169 = 13
- 5 × 8² × 3³ = 5 × 64 × 27 = 5 × 1728 = 8640
- ∛64 = 4
- 2⁵ = 32
Exam Special Tips
- Memorise square numbers up to 15² and cubes for 1, 2, 3, 4, 5, 10 — these are often non-calculator questions.
- Fractional powers save time: 27^(1/3) = ∛27 = 3.
- Always apply powers before multiplication unless brackets change order (BODMAS).
- Negative numbers under even roots are not real numbers.
Common Mistakes & Confusions
- Forgetting order of operations — writing √(a² + b²) instead of correctly squaring first.
- Mixing up 3² = 6 (wrong) instead of 9.
- Forgetting that cube roots can be negative: ∛(-8) = -2.
- Confusing powers and multiplication: 2³ × 2⁴ = 2⁷, not 2¹².
Written and Compiled By Sir Hunain Zia, World Record Holder With 154 Total A Grades, 7 Distinctions and 11 World Records For Educate A Change O Level And IGCSE Mathematics Full Scale Course