Indices
O Level and IGCSE Mathematics Cheat Sheet – 1.7 Indices I
What are Indices?
- Index (plural: indices) = power/exponent showing how many times to multiply a number by itself.
Example: 2³ = 2 × 2 × 2 = 8
| Type of Index | Example | Meaning |
|---|---|---|
| Positive | 5³ | 5 × 5 × 5 = 125 |
| Zero | a⁰ | Always = 1 (a ≠ 0) |
| Negative | 2⁻³ | 1 / (2³) = 1/8 |
| Fractional | 16^(1/2) | √16 = 4 |
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
Rules of Indices
| Rule | Statement | Example |
|---|---|---|
| 1. Multiplication (same base) | aᵐ × aⁿ = aᵐ⁺ⁿ | 2³ × 2⁴ = 2⁷ = 128 |
| 2. Division (same base) | aᵐ ÷ aⁿ = aᵐ⁻ⁿ | 5⁶ ÷ 5² = 5⁴ = 625 |
| 3. Power of a Power | (aᵐ)ⁿ = aᵐⁿ | (3²)³ = 3⁶ = 729 |
| 4. Power of a Product | (ab)ⁿ = aⁿbⁿ | (2×5)³ = 2³ × 5³ = 8 × 125 |
| 5. Power of a Fraction | (a/b)ⁿ = aⁿ/bⁿ | (3/4)² = 9/16 |
| 6. Negative Power | a⁻ⁿ = 1/aⁿ | 2⁻³ = 1/8 |
| 7. Fractional Power | aᵐ⁄ⁿ = ⁿ√(aᵐ) | 27^(2/3) = (³√27)² = 3² = 9 |
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
Worked Examples
- 2⁻³ × 2⁴ = 2¹ = 2
- (2³)² = 2⁶ = 64
- 2³ ÷ 2⁴ = 2⁻¹ = 1/2
- 8^(1/2) = √8
- 16^(3/4) = (⁴√16)³ = 2³ = 8
Special Values
- a⁰ = 1 (a ≠ 0)
- a¹ = a
- a^(-1) = reciprocal of a
- Square root: a^(1/2) = √a
- Cube root: a^(1/3) = ∛a
Exam Special Tips
- Always simplify using index rules first before calculating.
- Negative indices do not mean negative numbers — they mean reciprocals.
- For fractional indices: denominator = root, numerator = power.
- If bases are different, do not apply multiplication or division rules.
- Brackets matter: 2³² means (2³)², not 2³².
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
Common Mistakes & Confusions
- Thinking a⁰ = 0 (wrong — it’s 1).
- Forgetting to apply reciprocal for negative indices.
- Mixing order in fractional indices: a^(2/3) ≠ (a²)³.
- Applying rules to different bases (invalid).