AP PRECALCULUS — PREREQUISITE REVIEW
Exponential Functions and Rules of Exponents
Notes — Prerequisite Topic 6
💡 Learning Objectives
By the end of this lesson you will be able to:
- State and apply the product, quotient, and power rules of exponents
- Simplify expressions involving zero exponents, negative exponents, and rational exponents
- Recognize the form f(x) = a · bˣ and identify its key features
- Distinguish exponential growth (b > 1) from exponential decay (0 < b < 1)
- Compare the growth of an exponential function with that of a polynomial function
- Apply exponential models to compound interest, population, and decay problems
1. Rules of Exponents
Exponent rules are simply shorthand for repeated multiplication. Every rule below comes from writing a product out in full and counting factors.
The six core rules of exponents. Each rule assumes the base is a real number (nonzero where needed), and the exponents are real numbers.
Rule | Formula | Worked example |
|---|---|---|
Product rule | aᵐ · aⁿ = aᵐ⁺ⁿ | x⁴ · x³ = x⁷ |
Quotient rule | aᵐ / aⁿ = aᵐ⁻ⁿ | x⁷ / x² = x⁵ |
Power of a power | (aᵐ)ⁿ = aᵐⁿ | (x²)⁵ = x¹⁰ |
Power of a product | (ab)ⁿ = aⁿbⁿ | (2x)³ = 8x³ |
Zero exponent | a⁰ = 1, a ≠ 0 | 7⁰ = 1 |
Negative exponent | a⁻ⁿ = 1/aⁿ | x⁻³ = 1/x³ |
Rational exponent | a^(1/n) = ⁿ√a | 8^(1/3) = 2 |
General rational | a^(m/n) = ⁿ√(aᵐ) | 8^(2/3) = 4 |
⚠️ Common mistakes
1. (x + y)ⁿ ≠ xⁿ + yⁿ. There is no shortcut; you must expand the binomial.
2. aᵐ + aⁿ does not simplify to aᵐ⁺ⁿ. Exponent rules apply to multiplication, not addition.
3. A negative exponent does not make the value negative; it takes the reciprocal. For example, 2⁻³ = 1/8 (positive).
📘 Example — Putting the rules together
Simplify ( (3x²y⁻³)² · (xy⁴) ) / (x³y⁻²).
Numerator: (3x²y⁻³)² = 9x⁴y⁻⁶, then (9x⁴y⁻⁶)(xy⁴) = 9x⁵y⁻².
Divide: 9x⁵y⁻² / (x³y⁻²) = 9x² · y⁻²⁺² = 9x² · y⁰ = 9x².
2. Exponential Functions
An exponential function has the form:
f(x) = a · bˣ,
where a is a nonzero constant (the initial value) and b is a positive constant (the base), with b ≠ 1.
Three key exponential graphs. Every exponential curve of this family passes through (0, 1) when a = 1, is always positive, and has the x-axis (y = 0) as a horizontal asymptote.
Feature | Value / behaviour |
|---|---|
Domain | All real numbers (−∞, ∞) |
Range | (0, ∞) when a > 0; (−∞, 0) when a < 0 |
Horizontal asymptote | y = 0 |
y-intercept | (0, a) |
Growth if b > 1 | function increases; rises without bound as x → ∞ |
Decay if 0 < b < 1 | function decreases; approaches 0 as x → ∞ |
2.1 The number e
The number e ≈ 2.71828… is a universal constant that arises whenever growth is continuous rather than compounded at discrete intervals. The function f(x) = eˣ is called the natural exponential function and plays a starring role throughout AP Precalculus and beyond.
2.2 Growth vs. decay — quick test
- b > 1: exponential growth. The function is increasing.
- 0 < b < 1: exponential decay. The function is decreasing.
- b = 1: not exponential — the function is the constant a.
📘 Example — Growth or decay?
• f(x) = 3(1.08)ˣ has b = 1.08, so this is growth.
• g(x) = 100(0.85)ˣ has b = 0.85, so this is decay.
• h(x) = 2 · 5⁻ˣ. Rewrite as 2 · (1/5)ˣ. Base 1/5 < 1, so this is decay.
3. Transformations of Exponentials
The general exponential f(x) = a · bˣ⁻ʰ + k takes the parent b^x and:
- Stretches/reflects vertically by the factor a
- Shifts horizontally by h (right if h > 0, left if h < 0)
- Shifts vertically by k (up if k > 0, down if k < 0)
- Moves the horizontal asymptote from y = 0 to y = k
📘 Example — Identify the key features
For f(x) = 2 · 3^(x − 1) − 4:
Base b = 3 (growth), parent graph shifted 1 right and 4 down.
Horizontal asymptote: y = −4.
y-intercept: f(0) = 2 · 3⁻¹ − 4 = 2/3 − 4 = −10/3.
4. Comparing Exponential and Polynomial Growth
💡 Eventually, exponentials win
For any polynomial p(x) and any exponential f(x) = bˣ with b > 1, there exists some value of x beyond which f(x) > p(x) and the gap only widens. Exponentials eventually dominate polynomials — even the slow-looking 1.001ˣ will outpace x¹⁰⁰⁰ for large enough x.
5. Compound Interest and Natural Growth
💡 Two central formulas
• Compounded n times per year at rate r:
A = P · (1 + r/n)^(n·t).
• Continuously compounded:
A = P · e^(r·t).
Here P is the principal, r is the annual rate expressed as a decimal, and t is time in years.
📘 Example — Comparing compounding frequencies
You invest $1000 for 5 years at a nominal annual rate of 6%.
Annual compounding: 1000 · (1.06)⁵ ≈ $1338.23.
Monthly compounding: 1000 · (1 + 0.06/12)^60 ≈ $1348.85.
Continuous compounding: 1000 · e^(0.06 · 5) ≈ $1349.86.
📘 Example — Radioactive decay
A certain isotope decays so that 5% of the material disappears per hour. Starting with 80 g, the mass t hours later is m(t) = 80 · (0.95)ᵗ.
After 10 hours: m(10) = 80 · (0.95)¹⁰ ≈ 80 · 0.5987 ≈ 47.90 g.
6. Summary
- The exponent rules turn products into sums of exponents and quotients into differences
- Negative exponents are reciprocals, zero exponent gives 1, rational exponents are roots
- Exponential functions f(x) = a · bˣ are always positive (when a > 0), pass through (0, a), and have asymptote y = 0
- Growth when b > 1; decay when 0 < b < 1
- Use A = P(1 + r/n)^(nt) for discrete compounding and A = Pe^(rt) for continuous compounding