L34 Solutions — Index Laws and Scientific Notation

1. Index laws. Fluency

   • (a) x⁴ × x⁶ = x¹⁰.
   • (b) a⁹ ÷ a⁴ = a⁵.
   • (c) (m³)⁴ = m¹².
   • (d) (3b²)³ = 3³ × b⁶ = 27b⁶.

2. Negative and zero indices. Fluency

   • (a) 5⁰ = 1.
   • (b) 2⁻⁴ = 1/2⁴ = 1/16.
   • (c) 1/x⁵ = x⁻⁵.
   • (d) 6x⁻² × 2x³ = 12x⁻²⁺³ = 12x.

3. Fractional indices. Fluency

   • (a) 64^1/2 = √64 = 8.
   • (b) 125^1/3 = ∛125 = 5.
   • (c) 16^3/4 = (∜16)³ = 2³ = 8.
   • (d) 32^−2/5 = 1/(32^2/5) = 1/(⁵√32)² = 1/2² = 1/4.

4. Scientific notation. Fluency

   • (a) 7.25 × 10⁶.
   • (b) 9.4 × 10⁻⁵.
   • (c) 5.3 × 10⁴ = 53 000.
   • (d) (3 × 4) × 10⁴⁺³ = 12 × 10⁷ = 1.2 × 10⁸.

5. Exponential growth pattern. Understanding

   • (a) 2⁰ = 1. The graph crosses the y-axis at y = 1.
   • (b) 2^3.5 = 2^7/2 = √128 ≈ 11.3. (Halfway between 8 and 16.)
   • (c) 2^x > 0 for all x — the exponential function is always positive. No matter how negative x becomes, 2^x approaches zero but never reaches it.
   • (d) When x = 0: y = 2⁰ = 1. So the y-intercept is 1, which follows directly from the zero index law a⁰ = 1.

6. Mixed simplification. Understanding

   • (a) (x³y²)² × x⁻² = x⁶y⁴ × x⁻² = x⁴y⁴.
   • (b) (4a³) ÷ (2a⁵) = 2a³⁻⁵ = 2a⁻² = 2/a².
   • (c) √(x⁵) = (x⁵)^1/2 = x^5/2.
   • (d) (2x²)³ ÷ (4x³) = 8x⁶ ÷ 4x³ = 2x³.

7. Scientific notation in context. Understanding

   • (a) 1.67 × 10⁻²⁷ = 0.000 000 000 000 000 000 000 000 001 67 kg (27 decimal places).
   • (b) 1.5 × 10⁸ km.
   • (c) 2.5 × 10⁹ × 4 = 10 × 10⁹ = 1.0 × 10¹⁰ operations.
   • (d) (4.2 + 1.8) × 10⁻⁶ = 6.0 × 10⁻⁶ g.

8. Combining index laws. Understanding

   • (a) 2⁵ × 2⁻³ ÷ 2⁰ = 2⁵⁻³⁻⁰ = 2² = 4.
   • (b) (27x³)^1/3 = 27^1/3 × x^3×1/3 = 3 × x = 3x.
   • (c) (x²y⁻¹)³ ÷ (x³y²) = x⁶y⁻³ ÷ x³y² = x³y⁻⁵ = x³/y⁵.
   • (d) a^m/n = a^{m × (1/n)} = (a^1/n)^m by the power-of-a-power law (a^p)^q = a^pq, with p = 1/n and q = m. ✓

9. Doubling and half-life. Problem Solving

   • (a) N(t) = 500 × 2^t.
   • (b) N(5) = 500 × 2⁵ = 500 × 32 = 16 000 cells.
   • (c) 16 000 = 1.6 × 10⁴.
   • (d) M(t) = 800 × (1/2)^t/6. After 18 hours: M(18) = 800 × (1/2)³ = 800 × 1/8 = 100 mg.

10. Index equation solving and estimation. Problem Solving

    • (a) 2^x = 64 = 2⁶ ⇒ x = 6.
    • (b) 3^x = 1/27 = 3⁻³ ⇒ x = −3.
    • (c) (2x)² = 144 ⇒ 2x = 12 ⇒ x = 6.
    • (d) Side = 200^1/2 = √200 ≈ 14.14 cm.