Topic A: Laws of Exponents
Lessons 1–4 · Product, quotient, power, and negative exponent rules.
01–02 — Lessons 1–2
Exponent Rules
- Product rule: aᵐ · aⁿ = aᵐ⁺ⁿ
- Quotient rule: aᵐ / aⁿ = aᵐ⁻ⁿ
- Power rule: (aᵐ)ⁿ = aᵐⁿ
- Zero exponent: a⁰ = 1 (a ≠ 0)
- Negative exponent: a⁻ⁿ = 1/aⁿ
- Fraction base: (a/b)ⁿ = aⁿ/bⁿ
x³ · x⁵ = x⁸ · (2x²)³ = 8x⁶ · x⁻² = 1/x²
Practice — Lessons 1–2
Q1 · Short Answer
Simplify: x⁴ · x⁷. Write as xⁿ.
Q2 · Multiple Choice
What is 5⁰?
03–04 — Lessons 3–4
Scientific Notation
- Scientific notation: a × 10ⁿ, where 1 ≤ a < 10.
- Large numbers: move decimal left → positive exponent.
- Small numbers: move decimal right → negative exponent.
- Multiply: multiply the a-values, add the exponents.
- Divide: divide the a-values, subtract the exponents.
0.000045 = 4.5 × 10⁻⁵ · (3×10⁴)(2×10³) = 6×10⁷
Practice — Lessons 3–4
Q3 · Multiple Choice
Write 67,000 in scientific notation.
Q4 · Short Answer
(4 × 10³) × (2 × 10⁵) = ? Write in scientific notation.
Topic B: Exponential Functions
Lessons 5–9 · Graphing, growth, decay, and applications.
05–07 — Lessons 5–7
Exponential Growth & Decay
- General form: f(x) = a · bˣ
- a: initial value (y-intercept). b: base (growth/decay factor).
- b > 1: exponential growth. 0 < b < 1: exponential decay.
- Percent change: growth rate r means b = 1 + r. Decay: b = 1 − r.
- No x-intercept (for a > 0). Asymptote at y = 0 (for standard exponential).
Population decreases 15%/yr from 1000: f(t) = 1000·(0.85)ᵗ. After 4 yr: f(4) ≈ 522.
Practice — Lessons 5–7
Q5 · Multiple Choice
A quantity grows by 20% each year starting at 500. What is the base b?
Q6 · Short Answer
Using f(t) = 400·(0.5)ᵗ, what is f(2)?
08–09 — Lessons 8–9
Geometric Sequences & Comparing Function Types
- Geometric sequence: multiply by constant ratio r each term.
- aₙ = a₁ · r^(n−1)
- vs. arithmetic sequence: arithmetic adds a constant difference.
- Geometric sequences are modeled by exponential functions.
- Comparing: exponential eventually overtakes linear and quadratic.
Sequence: 3, 6, 12, 24 → r = 2 → a₅ = 3·2⁴ = 48
Practice — Lessons 8–9
Q7 · Multiple Choice
Which sequence is geometric?
Q8 · Short Answer
Find the 5th term of the sequence 2, 6, 18, 54, …