Indices And Surds (Copy)
INDICES AND SURDS — O LEVEL & IGCSE ADDITIONAL MATHEMATICS FORMULA SHEET + COMMON MISTAKES
| Topic | Formula / Rule | Common Mistake |
|---|---|---|
| Multiplication Rule | aᵐ × aⁿ = aᵐ⁺ⁿ | Multiplying powers instead of adding |
| Division Rule | aᵐ/aⁿ = aᵐ⁻ⁿ | Subtracting wrong way round |
| Power of Power | (aᵐ)ⁿ = aᵐⁿ | Adding instead of multiplying |
| Power of Product | (ab)ⁿ = aⁿbⁿ | Applying power to one term only |
| Power of Fraction | (a/b)ⁿ = aⁿ/bⁿ | Squaring numerator only |
| Zero Index | a⁰ = 1 | Writing 0 |
| Negative Index | a⁻ⁿ = 1/aⁿ | Keeping negative powers in final answer |
| Fractional Index | a¹⁄² = √a | Writing a/2 |
| Cube Root Index | a¹⁄³ = ∛a | Confusing with square root |
| General Fractional Index | aᵐ⁄ⁿ = ⁿ√(aᵐ) | Wrong root/power order |
| Scientific Notation | a × 10ⁿ | Number before decimal not between 1 and 10 |
| Large Number Standard Form | Positive power of 10 | Using negative powers |
| Small Number Standard Form | Negative power of 10 | Using positive powers |
| Surd Definition | Irrational root | Trying to decimalise exact answers |
| Simplifying Surds | √(ab) = √a × √b | Splitting non-perfect squares wrongly |
| Adding Surds | a√x + b√x = (a + b)√x | Adding unlike surds |
| Multiplying Surds | √a × √b = √ab | Adding instead of multiplying |
| Rationalising Denominator | Remove surd from denominator | Leaving surd below line |
| Conjugates | a + √b and a − √b | Same signs instead of opposite |
| Difference of Squares | (a + b)(a − b) = a² − b² | Forgetting middle terms cancel |
| Exact Values | Leave answers in surd form | Giving decimal approximation |
| Index Equation | Make bases equal | Solving directly without converting |
| Negative Powers | Convert to reciprocal | Leaving negative indices |
| Expanding Surds | Multiply every term | Missing middle terms |
| Squaring Surds | (√a)² = a | Keeping square root |
| Root Restriction | √x only if x ≥ 0 | Allowing negative roots |
| Surd Simplification | √12 = 2√3 | Writing √12 unchanged |
| Like Surds | 3√2 + 5√2 = 8√2 | Writing 8√4 |
| Unlike Surds | √2 + √3 cannot simplify | Combining incorrectly |
| Rationalising Single Surd | 1/√a = √a/a | Multiplying one side only |
| Rationalising Binomial | Multiply by conjugate | Wrong conjugate used |
| Standard Form Multiplication | Multiply numbers and powers separately | Adding everything together |
| Standard Form Division | Divide numbers and subtract powers | Adding powers instead |
Written and Compiled By Sir Hunain Zia (AYLOTI), World Record Holder With 154 Total A Grades, 11 World Records and 7 Distinctions, Educate A Change.
QUICK INDEX RULES TABLE
| Rule | Formula |
|---|---|
| Multiplication | aᵐ × aⁿ = aᵐ⁺ⁿ |
| Division | aᵐ/aⁿ = aᵐ⁻ⁿ |
| Power of Power | (aᵐ)ⁿ = aᵐⁿ |
| Power of Product | (ab)ⁿ = aⁿbⁿ |
| Power of Fraction | (a/b)ⁿ = aⁿ/bⁿ |
| Zero Index | a⁰ = 1 |
| Negative Index | a⁻ⁿ = 1/aⁿ |
| Fractional Index | a¹⁄ⁿ = ⁿ√a |
QUICK SURD RULES TABLE
| Rule | Formula |
|---|---|
| Product Rule | √a × √b = √ab |
| Quotient Rule | √a/√b = √(a/b) |
| Rationalising | 1/√a = √a/a |
| Conjugates | (a + b)(a − b) = a² − b² |
| Square Root | √a² = a |
| Cube Root | ∛a³ = a |
Written and Compiled By Sir Hunain Zia (AYLOTI), World Record Holder With 154 Total A Grades, 11 World Records and 7 Distinctions, Educate A Change.
COMMON SURD SIMPLIFICATIONS
| Expression | Simplified Form |
|---|---|
| √8 | 2√2 |
| √12 | 2√3 |
| √18 | 3√2 |
| √20 | 2√5 |
| √27 | 3√3 |
| √45 | 3√5 |
| √50 | 5√2 |
| √72 | 6√2 |
COMMON EXACT VALUES
| Angle | Value |
|---|---|
| sin30° | 1/2 |
| cos30° | √3/2 |
| tan30° | 1/√3 |
| sin45° | √2/2 |
| cos45° | √2/2 |
| tan45° | 1 |
| sin60° | √3/2 |
| cos60° | 1/2 |
| tan60° | √3 |
EXAM HACKS
| Situation | Fast Trick |
|---|---|
| Negative index | Move across fraction line |
| Fractional index | Denominator = root |
| Standard form | First number between 1 and 10 |
| Surd simplification | Find largest square factor |
| Rationalising | Multiply by conjugate |
| Like surds | Combine coefficients only |
| Index equations | Convert to same base |
| Exact values | Keep surd form |
Written and Compiled By Sir Hunain Zia (AYLOTI), World Record Holder With 154 Total A Grades, 11 World Records and 7 Distinctions, Educate A Change.
MOST COMMON LOSSES OF MARKS
| Mistake | Why Marks Are Lost |
|---|---|
| Wrong index law | Entire simplification incorrect |
| Adding unlike surds | Invalid algebra |
| Forgetting reciprocal | Negative powers remain |
| Decimalising surds | Exact-answer marks lost |
| Incorrect rationalisation | Final answer not simplified |
| Wrong conjugate | Expansion becomes incorrect |
| Poor standard form | Scientific notation invalid |
| Forgetting brackets | Wrong powers |
| Expanding incorrectly | Missing terms |
| Leaving answer unsimplified | Final accuracy marks lost |
Written and Compiled By Sir Hunain Zia (AYLOTI), World Record Holder With 154 Total A Grades, 11 World Records and 7 Distinctions, Educate A Change.