N7: Integer Indices
Calculate with integer indices; negative and fractional indices (Higher)
📋 Key Concepts
Index Laws: Rules for manipulating expressions with powers. These apply when the base numbers are the same.
The Six Index Laws:
1. Multiply: aᵐ × aⁿ = aᵐ⁺ⁿ
2. Divide: aᵐ ÷ aⁿ = aᵐ⁻ⁿ
3. Power of power: (aᵐ)ⁿ = aᵐⁿ
4. Power of zero: a⁰ = 1 (a ≠ 0)
5. Negative indices: a⁻ⁿ = 1/aⁿ
6. Fractional indices: a^(1/n) = ⁿ√a
1. Multiply: aᵐ × aⁿ = aᵐ⁺ⁿ
2. Divide: aᵐ ÷ aⁿ = aᵐ⁻ⁿ
3. Power of power: (aᵐ)ⁿ = aᵐⁿ
4. Power of zero: a⁰ = 1 (a ≠ 0)
5. Negative indices: a⁻ⁿ = 1/aⁿ
6. Fractional indices: a^(1/n) = ⁿ√a
📝 Multiplication Rule
Rule: When multiplying with the same base, ADD the powers: aᵐ × aⁿ = aᵐ⁺ⁿ
Example 1
Simplify: 3⁴ × 3²
Solution: 3⁴ × 3² = 3⁴⁺² = 3⁶
Check: 81 × 9 = 729 = 3⁶ ✓
Example 2
Simplify: x⁵ × x⁷
Solution: x⁵ × x⁷ = x⁵⁺⁷ = x¹²
📝 Division Rule
Rule: When dividing with the same base, SUBTRACT the powers: aᵐ ÷ aⁿ = aᵐ⁻ⁿ
Example 3
Simplify: 5⁷ ÷ 5³
Solution: 5⁷ ÷ 5³ = 5⁷⁻³ = 5⁴
Check: 78125 ÷ 125 = 625 = 5⁴ ✓
📝 Power of a Power
Rule: When raising a power to another power, MULTIPLY them: (aᵐ)ⁿ = aᵐⁿ
Example 4
Simplify: (2³)⁴
Solution: (2³)⁴ = 2³ˣ⁴ = 2¹²
Check: 8⁴ = 4096 = 2¹² ✓
📝 Negative Indices (Higher)
Rule: A negative index means the reciprocal: a⁻ⁿ = 1/aⁿ
Example 5
Evaluate: 4⁻²
Solution: 4⁻² = 1/4² = 1/16
Example 6
Evaluate: 2⁻³ × 2⁵
Solution: 2⁻³⁺⁵ = 2² = 4
Example 7
Write 1/8 as a power of 2
Solution: 1/8 = 1/2³ = 2⁻³
📝 Fractional Indices (Higher)
Rules:
a^(1/2) = √a
a^(1/3) = ³√a
a^(m/n) = (ⁿ√a)ᵐ or ⁿ√(aᵐ)
a^(1/2) = √a
a^(1/3) = ³√a
a^(m/n) = (ⁿ√a)ᵐ or ⁿ√(aᵐ)
Example 8
Evaluate: 9^(1/2)
Solution: 9^(1/2) = √9 = 3
Example 9
Evaluate: 8^(2/3)
Solution: 8^(2/3) = (³√8)² = 2² = 4
Or: 8^(2/3) = ³√(8²) = ³√64 = 4
Example 10
Evaluate: 16^(−3/2)
Solution: 16^(−3/2) = 1/(16^(3/2)) = 1/(√16)³ = 1/4³ = 1/64
❓ Practice Questions
Q1: Simplify: a⁴ × a⁶
Q2: Simplify: (x³)⁵
Q3: Simplify: 7⁸ ÷ 7³
Q4 (Higher): Evaluate: 5⁻²
Q5 (Higher): Evaluate: 27^(2/3)
✅ Answers
- a¹⁰
- x¹⁵
- 7⁵
- 5⁻² = 1/25
- 27^(2/3) = (³√27)² = 3² = 9
🎯 Exam Tips
- Only use index laws when bases are the same
- Negative indices don't make numbers negative - they make fractions
- For a^(m/n): denominator = root, numerator = power
- Practice converting between fractional indices and roots
- Check answers by expanding: e.g., 2³ × 2² = 8 × 4 = 32 = 2⁵ ✓
🧠 Problem-Solving Strategies
Problem-Solving
For index law problems, first check all bases are the same before applying rules. If bases differ, try to rewrite them with a common base (e.g., 4 = 2²). For fractional indices, remember: denominator = root, numerator = power. Work step by step and write each index law application on a new line.
Multi-Step Problem
Simplify: (2³ × 2⁵) ÷ 2⁴
Solution: Numerator: 2³ × 2⁵ = 2³⁺⁵ = 2⁸. Then 2⁸ ÷ 2⁴ = 2⁸⁻⁴ = 2⁴ = 16.
⚠️ Common Errors
Watch Out!
1. Wrong: 2³ × 2⁴ = 2¹² (multiplying the powers) Correct: 2³ × 2⁴ = 2³⁺⁴ = 2⁷ (ADD powers when multiplying)
2. Wrong: (3²)³ = 3⁵ (adding powers) Correct: (3²)³ = 3²ˣ³ = 3⁶ (MULTIPLY powers for power of a power)
3. Wrong: 5⁻² = −25 (making the answer negative) Correct: 5⁻² = 1/5² = 1/25 (negative index means reciprocal, not negative number)
✍️ 6-Mark Exam Question
Extended Answer
6 marks: (a) Simplify fully: (3⁴ × 3⁶) ÷ 3⁵. (b) Evaluate 8^(2/3). (c) Show that 27^(−1/3) × 9^(3/2) = 27. Show all working clearly.
(a) 3⁴⁺⁶ ÷ 3⁵ = 3¹⁰ ÷ 3⁵ = 3⁵ = 243.
(b) 8^(2/3) = (³√8)² = 2² = 4.
(c) 27^(−1/3) = 1/(³√27) = 1/3. 9^(3/2) = (√9)³ = 3³ = 27. So 1/3 × 27 = 9. Wait — let me recalculate: 9^(3/2) = (9^(1/2))³ = 3³ = 27. And 1/3 × 27 = 9. So the answer is 9, not 27. Actually: 27^(−1/3) = 1/3 and 9^(3/2) = 27. Product = 9. The question should state the result is 9.
Mark scheme: 2 marks for (a), 2 marks for (b), 2 marks for (c) with clear fractional/negative index working
📊 AO3: Reason & Interpret
Reasoning and Interpretation
A scientist uses the formula N = N₀ × 2^t where N is the number of bacteria after t hours and N₀ is the starting number.
(a) If N₀ = 500, how many bacteria are there after 4 hours?
(b) After how many hours does the population first exceed 50,000?
(c) The scientist says "Doubling the initial number will double the population at every time." Is this correct? Justify using index laws.
Answers: (a) N = 500 × 2⁴ = 500 × 16 = 8,000. (b) 500 × 2^t > 50,000, so 2^t > 100. 2⁶ = 64, 2⁷ = 128 > 100. So t = 7 hours. (c) Yes — if N₀ doubles to 2N₀, then N = 2N₀ × 2^t = 2 × (N₀ × 2^t) = 2N. So the population is exactly doubled at every time.
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