📊 Lesson 6: Exponents & Radicals

Power rules, scientific notation, and roots — the language of scaling laws

🎯 Learning Objectives

• Apply exponent rules fluently: product, quotient, power of power, negative, fractional
• Convert between scientific notation and decimal form
• Simplify radical expressions and rationalize denominators
• Handle units with exponents (m², m³, m/s², etc.)

🔑 Key Concepts

Exponent Rules (Memorize These!)

Rule	Formula	Example
Product	xᵃ·xᵇ = xᵃ⁺ᵇ	x²·x³ = x⁵
Quotient	xᵃ/xᵇ = xᵃ⁻ᵇ	x⁵/x² = x³
Power of Power	(xᵃ)ᵇ = xᵃᵇ	(x²)³ = x⁶
Power of Product	(xy)ᵃ = xᵃyᵃ	(2x)³ = 8x³
Power of Quotient	(x/y)ᵃ = xᵃ/yᵃ	(x/y)² = x²/y²
Zero Exponent	x⁰ = 1 (x≠0)	5⁰ = 1
Negative Exponent	x⁻ᵃ = 1/xᵃ	x⁻² = 1/x²
Fractional Exponent	x^(m/n) = n√(xᵐ) = (n√x)ᵐ	x^(1/2) = √x, x^(3/2) = √(x³)

Scientific Notation

Physics deals with huge and tiny numbers: speed of light = 3.00×10⁸ m/s, electron charge = 1.602×10⁻¹⁹ C.

Format: a × 10ⁿ where 1 ≤ |a| < 10 and n is an integer.

Operation	Rule	Example
Multiply	Multiply coefficients, add exponents	(2×10³)(3×10⁴) = 6×10⁷
Divide	Divide coefficients, subtract exponents	(8×10⁶)/(2×10²) = 4×10⁴
Power	Raise coefficient, multiply exponent	(2×10³)² = 4×10⁶

Radicals

√x means the non-negative number whose square is x.

• √(ab) = √a · √b (for a,b ≥ 0)
• √(a/b) = √a / √b (for a ≥ 0, b > 0)
• √(a²) = |a| (absolute value!)

📐 Worked Examples

Example 1: Simplifying with Exponents

Simplify: (2×10³)(5×10⁻²) / (10⁴)

= (10×10¹) / 10⁴
= 10² / 10⁴
= 10⁻²
= 0.01
            

Example 2: Fractional Exponents in Physics

Period of pendulum: T = 2π√(L/g). Rewrite with fractional exponents.

T = 2π (L/g)^(1/2)
T = 2π L^(1/2) g^(-1/2)
            

Example 3: Units with Exponents

Simplify: (m/s)² · (s³/m)

= m²/s² · s³/m
= m²⁻¹ · s³⁻²
= m¹ · s¹
= m·s
            

🏋️ Practice Problems

1. Simplify: x³·x⁵, x⁸/x³, (x²)⁴, (2x³)²
2. Write in scientific notation: 450,000; 0.0000032; 6.02×10²³ × 2×10⁴
3. Simplify: (3×10⁴)², (6×10⁻³)/(2×10⁻⁷), √(4×10⁶)
4. Simplify: √50, √72, √(18x⁴), √(x⁶y⁸)
5. Rationalize: 1/√2, 3/√5, √3/√2
6. Gravitational force: F = Gm₁m₂/r². If r doubles, what happens to F? (inverse square law)
7. Kinetic energy: KE ∝ v². If v triples, KE increases by what factor?
8. Simplify units: (kg·m/s²)·m / (kg·m²/s³) = ?
9. Universal gravitation: F ∝ 1/r². Express as fractional exponent.

✅ Answers

Click to reveal answers

1) x⁸, x⁵, x⁸, 4x⁶ 2) 4.5×10⁵, 3.2×10⁻⁶, 1.2×10²⁸ 3) 9×10⁸, 3×10⁴, 2×10³

4) 5√2, 6√2, 3x²√2, x³y⁴ 5) √2/2, 3√5/5, √6/2 6) F becomes ¼ (inverse square)

7) 9× (v² relation) 8) s 9) F ∝ r⁻²

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