Exponents and Exponential Functions

This page covers the fundamental concepts of exponents and exponential functions, including exponent rules and examples with answers, as well as the characteristics of exponential growth and decay.

Exponent Rules

The page begins by listing several important exponent rules:

1. Zero Rule: a⁰ = 1
2. Product Rule: aᵐ × aⁿ = aᵐ⁺ⁿ
3. Quotient Rule: aᵐ ÷ aⁿ = aᵐ⁻ⁿ
4. Power of a Product: (ab)ᵐ = aᵐbᵐ
5. Power of a Quotient: $a/b$ᵐ = aᵐ/bᵐ
6. Power of a Power: (aᵐ)ⁿ = aᵐⁿ
7. Negative Exponent: a⁻ᵐ = 1/aᵐ
8. Fractional Exponent: a^$m/n$ = ⁿ√(aᵐ)

Definition: An exponential function is a function where the base is a constant and the exponent is a variable, typically expressed as y = abˣ.

Exponential Growth vs. Decay

The page then discusses the concepts of exponential growth and decay:

Highlight: For a positive base a, the function y = abˣ can be classified as either exponential growth or exponential decay, depending on the value of b.

• If b > 1, the function represents exponential growth.
• If 0 < b < 1, the function represents exponential decay.
• If a is negative, the function is neither exponential growth nor decay.

Example: Examples of exponential growth include y = 2ˣ, y = 3(4)ˣ, and y = (3/2)ˣ.

Example: Examples of exponential decay include y = (1/2)ˣ, y = (0.5)ˣ, and y = 2(1/3)ˣ.

Example: Examples that are neither growth nor decay include y = -2(3)ˣ, y = -(0.33)ˣ, and y = -5(3)⁻ˣ.

This comprehensive overview provides students with a solid foundation in exponent rules and examples, as well as the concepts of exponential growth and decay, which are crucial for understanding more advanced topics in mathematics and science.