Exponents & Exponential Functions

The laws of exponents, scientific notation, and modeling growth and decay.

In this module

Laws of Exponents

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.
Power of a product: (ab)ⁿ = aⁿbⁿ.

Worked example

Simplify (2x³)(3x⁵) / x².

Solution. Multiply the numerator: 2·3 = 6, x³·x⁵ = x⁸, giving 6x⁸. Divide: x⁸/x² = x⁶. Result: 6x⁶.

A number in scientific notation is written as a × 10ⁿ, where 1 ≤ a < 10.
To multiply or divide numbers in scientific notation, multiply/divide the "a" parts and add/subtract the exponents.
Converting a standard number: count how many places the decimal moves; moving left gives a positive exponent, moving right gives a negative one.

Worked example

Multiply (3 × 10⁴)(2 × 10⁻⁶).

Solution. Multiply the coefficients: 3 × 2 = 6. Add the exponents: 4 + (−6) = −2. Result: 6 × 10⁻².

General model: y = a(1 + r)ᵗ for growth or y = a(1 − r)ᵗ for decay, where a is the initial amount, r is the rate, and t is time.
Growth factors (1 + r) are greater than 1; decay factors (1 − r) are between 0 and 1.

Worked example

A population of 500 grows 4% per year. Write a model and find the population after 3 years.

Solution. Model: y = 500(1.04)ᵗ. After 3 years: y = 500(1.04)³ ≈ 500(1.1249) ≈ 562.

The graph of y = a·bˣ has a horizontal asymptote at y = 0 and never crosses the x-axis.
If b > 1, the function increases (growth); if 0 < b < 1, it decreases (decay).
The y-intercept of y = a·bˣ is always (0, a), since b⁰ = 1.

Exponential growth vs. decay

y = 2ˣ grows and y = (½)ˣ decays. Both pass through (0, 1), and both approach the x-axis (y = 0) without ever reaching it.

Worked example

Describe the graph of y = 3(0.5)ˣ.

Solution. y-intercept (0, 3); since b = 0.5 is between 0 and 1, the graph decays, approaching but never touching y = 0 as x increases.

2 of this module's 5 quiz questions. Pick your answers, then check them in the app, with an explanation for every question.

1. Simplify: x⁵ · x³.

2x⁸x²x⁸x¹⁵

2. Simplify: (2a²)³.

8a⁵6a⁶2a⁶8a⁶