Chapter 12
Rules of Exponents & Large-Number Arithmetic
Powers of 10, scientific notation, compound growth, unit scaling
12.1 Powers of 10
10¹ = 10 10⁶ = 1 000 000 (one million) 10² = 100 10⁹ = 1 000 000 000 (one billion) 10³ = 1 000 10¹² = one trillion
12.2 The seven core rules
Rule 1 Product: aᵐ × aⁿ = aᵐ⁺ⁿ Rule 2 Quotient: aᵐ ÷ aⁿ = aᵐ⁻ⁿ Rule 3 Power of a power: (aᵐ)ⁿ = aᵐˣⁿ Rule 4 Power of product: (ab)ⁿ = aⁿ × bⁿ Rule 5 Zero exponent: a⁰ = 1 Rule 6 Negative exp: a⁻ⁿ = 1 ÷ aⁿ Rule 7 Fractional exp: a¹/ⁿ = nth root of a
12.3 Scientific notation
number = M × 10ⁿ where 1 ≤ M < 10
Examples: 43 500 000 = 4.35 × 10⁷. 0.0072 = 7.2 × 10⁻³. 630 000 000 000 = 6.3 × 10¹¹.
12.4 Multiplying and dividing in scientific notation
Multiply: (M1 × 10ᵃ) × (M2 × 10ᵇ) = (M1 × M2) × 10ᵃ⁺ᵇ Divide: (M1 × 10ᵃ) ÷ (M2 × 10ᵇ) = (M1 ÷ M2) × 10ᵃ⁻ᵇ
To ADD or SUBTRACT: align exponents first. 5.1 × 10⁸ + 7.3 × 10⁷ = 5.1 × 10⁸ + 0.73 × 10⁸ = 5.83 × 10⁸.
12.5 Unit scaling
1 km = 10³ m → 1 km² = 10⁶ m² 1 tonne = 10³ kg 1 megatonne = 10⁹ kg
12.6 Compound growth and the Rule of 70
Value after n years = Start × (1 + r/100)ⁿ
Example: €5 000 at 4%/year for 10 years = €7 401. Rule of 70: doubling time ≈ 70 ÷ r years. At 5%, doubles in ~14 years.
12.7 Critical errors to avoid
💡 aᵐ + aⁿ ≠ aᵐ⁺ⁿ. The product rule applies only to multiplication: 3² + 3³ = 9 + 27 = 36, not 3⁵ = 243.
💡 (a + b)ⁿ ≠ aⁿ + bⁿ. Example: (5+2)² = 49, not 25+4 = 29.
💡 1 million = 10⁶. 1 billion = 10⁹. 1 trillion = 10¹². Confusing million and billion is a 1 000× error — very common in GDP questions.