Solutions — Indices and Index Laws

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1. Q1 — Evaluating expressions

   (a) 2⁵ = 32   (b) 3⁴ = 81   (c) 5⁰ = 1   (d) 4⁻² = 1/4² = 1/16   (e) 8^1/3 = ³√8 = 2   (f) 16^3/4 = (⁴√16)³ = 2³ = 8

2. Q2 — Multiplication law

   (a) 5³ × 5⁴ = 5⁷ = 78 125   (b) x⁷ × x² = x⁹   (c) 3a⁴ × 4a³ = 12a⁷ = 12a⁷   (d) 2m⁵ × 3m⁻² = 6m³ = 6m³

3. Q3 — Division law

   (a) 6⁸ ÷ 6³ = 6⁵ = 7776   (b) x⁹ ÷ x⁴ = x⁵   (c) 12a⁶ ÷ 4a² = 3a⁴   (d) 15x³ ÷ 5x⁵ = 3x⁻² = 3/x²

4. Q4 — Power of a power law

   (a) (2³)⁴ = 2¹² = 4096   (b) (x⁵)² = x¹⁰   (c) (3a²)³ = 27a⁶ = 27a⁶   (d) (2x³y²)⁴ = 2⁴x¹²y⁸ = 16x¹²y⁸

5. Q5 — Scientific notation

   (a) 3 800 000 = 3.8 × 10⁶   (b) 0.000 072 = 7.2 × 10⁻⁵   (c) 450 000 000 = 4.5 × 10⁸   (d) 0.0000035 = 3.5 × 10⁻⁶

6. Q6 — Simplifying with positive indices

   (a) (x²y³)² × x⁻³y = x⁴y⁶ × x⁻³y = x¹y⁷ = xy⁷   (b) (4a³b⁻²) ÷ (2a⁻¹b³) = 2a³⁻⁽⁻¹⁾b⁻²⁻³ = 2a⁴b⁻⁵ = 2a⁴/b⁵   (c) (3x²)³ ÷ 9x⁴ = 27x⁶ ÷ 9x⁴ = 3x²   (d) (m⁻¹n²)³ × (m⁴n⁻³)² = m⁻³n⁶ × m⁸n⁻⁶ = m⁵n⁰ = m⁵

7. Q7 — Expressing as a single index

   (a) ⁴√(x³) = x^3/4   (b) 1/(³√x) = x^−1/3 = x^−1/3   (c) (√x)⁵ = (x^1/2)⁵ = x^5/2   (d) ³√(8x⁶) = (8)^1/3(x⁶)^1/3 = 2x² = 2x²

8. Q8 — Solving by matching bases

   (a) 3^x = 81 = 3⁴ → x = 4   (b) 2^x+1 = 16 = 2⁴ → x+1 = 4 → x = 3   (c) 5^2x−1 = 25 = 5² → 2x−1 = 2 → 2x = 3 → x = 3/2   (d) 4^x = 8 → (2²)^x = 2³ → 2^2x = 2³ → 2x = 3 → x = 3/2

9. Q9 — Scientific notation operations

   (a) (3 × 10⁴) × (2 × 10³) = 6 × 10⁷ = 6 × 10⁷   (b) (8 × 10⁷) ÷ (4 × 10³) = 2 × 10⁴ = 2 × 10⁴   (c) (5 × 10⁻³) × (6 × 10⁵) = 30 × 10² = 3 × 10³   (d) Time = distance ÷ speed = (1.5 × 10¹¹) ÷ (3 × 10⁸) = 0.5 × 10³ = 500 seconds

10. Q10 — Complex simplification

    (a) Numerator: (2x³y⁻¹)² = 4x⁶y⁻². Times 4x⁻¹y³ = 16x⁵y¹. Divide by 8x²y⁴: = 2x³y⁻³ = 2x³/y³   (b) (27a⁶b³)^2/3 = 27^2/3a⁴b² = 9a⁴b². Divide by 3a²b = 3a²b   (c) (a^m+n × a^m−n) / a^2m = a^{(m+n)+(m−n)} / a^2m = a^2m / a^2m = a⁰ = 1  ✓

11. Q11 — Indices in context

    (a) N = 500 × 2ⁿ. After 8 hours: N = 500 × 2⁸ = 500 × 256 = 128 000 bacteria.   (b) (1.67 × 10⁻²⁷) ÷ (9.11 × 10⁻³¹) = (1.67/9.11) × 10^{−27−(−31)} = 0.1833... × 10⁴ ≈ 1.83 × 10³. A proton is approximately 1830 times heavier.   (c) 4ⁿ = 1024 = 4⁵ → n = 5. Then 4⁵ × 8^5/3 = 1024 × (2³)^5/3 = 1024 × 2⁵ = 1024 × 32 = 32 768.