Indices II
2.4 Indices II – Cheat Sheet
1. Understanding Indices
- Index (exponent/power) tells you how many times to multiply the base by itself.
- Example: 2³ = 2 × 2 × 2 = 8
| Type | Example | Meaning |
|---|---|---|
| Positive index | 4³ | 4 × 4 × 4 = 64 |
| Zero index | a⁰ | 1 (a ≠ 0) |
| Negative index | a⁻³ | 1 / a³ |
| Fractional index | a¹ᐟ² | √a |
| Fractional negative index | a⁻¹ᐟ² | 1 / √a |
2. Rules of Indices
| Rule | Name | Example | Result |
|---|---|---|---|
| aᵐ × aⁿ = aᵐ⁺ⁿ | Multiplication | 2³ × 2⁴ | 2⁷ |
| aᵐ ÷ aⁿ = aᵐ⁻ⁿ | Division | 5⁶ ÷ 5² | 5⁴ |
| (aᵐ)ⁿ = aᵐⁿ | Power of a power | (3²)³ | 3⁶ |
| (ab)ⁿ = aⁿbⁿ | Power of a product | (2x)³ | 2³x³ = 8x³ |
| (a/b)ⁿ = aⁿ / bⁿ | Power of a quotient | (x/2)² | x² / 4 |
3. Working with Zero, Negative, and Fractional Indices
| Type | Example | Result |
|---|---|---|
| Zero index | 7⁰ | 1 |
| Negative index | 2⁻³ | 1 / 8 |
| Fractional index | 9¹ᐟ² | 3 |
| Fractional index | 27²ᐟ³ | (³√27)² = 3² = 9 |
4. Simplifying Expressions with Indices
Example 1:
3² × 3⁴ = 3⁶ = 729
Example 2:
x⁵ ÷ x² = x³
Example 3:
(2x³)² = 4x⁶
Example 4:
(27x⁶y³)¹ᐟ³ = 3x²y¹
5. Solving Equations with Indices
Case 1: Same base
- 3^(2x) = 9
- 9 = 3²
- 3^(2x) = 3² → 2x = 2 → x = 1
Case 2: Same base, variables on both sides
- 5^(x + 1) = 25^x
- 25 = 5²
- 5^(x + 1) = 5^(2x) → x + 1 = 2x → x = 1
6. Worked Examples from Syllabus
| Problem | Process | Answer |
|---|---|---|
| 3² × 3⁴ | Add powers | 3⁶ |
| x⁵ ÷ x² | Subtract powers | x³ |
| (2x³)² | Square coefficient, multiply powers | 4x⁶ |
| 3^(2x) = 2 | No logs needed, but not same base → leave in form | |
| 5^(x + 1) = 25^x | Convert 25 to 5², equate powers | x = 1 |
| 3⁻² | Negative index | 1/9 |
| 16¹ᐟ² | Fractional index | 4 |
| 8⁻²ᐟ³ | Negative fractional index | 1 / (³√8)² = 1/4 |