Overview
This lecture covers the fundamental properties of exponents, including how to simplify expressions using exponent rules for products, quotients, negative and zero exponents, and powers, as well as common mistakes to avoid.
Basics of Exponents
- An exponent shows repeated multiplication of a base: ( a^n = a \times a \times … \times a ) (n times).
- A base with no visible exponent has an exponent of 1.
- Multiplication is commutative; order doesn't matter.
Product Property of Exponents
- When multiplying like bases, add their exponents: ( a^m \times a^n = a^{m+n} ).
- Only add exponents for terms with the same base and multiplication, not addition.
- Cannot combine terms with different bases using exponent rules.
Quotient Property of Exponents
- When dividing like bases, subtract exponents: ( a^m / a^n = a^{m-n} ).
- Only applies when dividing terms with the same base.
- Addition/subtraction in numerator or denominator prevents simplification with exponent rules.
Negative and Zero Exponents
- Negative exponent: ( a^{-n} = 1/a^n ); move base across the fraction bar to make exponent positive.
- Zero exponent: Any nonzero base to the zero power equals 1; ( a^0 = 1 ).
Power Property of Exponents
- When raising an exponential to a power, multiply exponents: ( (a^m)^n = a^{mn} ).
Power of a Product and Quotient
- Power of a product: ( (ab)^n = a^n b^n ); distribute exponent to each factor inside.
- Power of a quotient: ( (a/b)^n = a^n / b^n ); distribute exponent to both numerator and denominator.
- Cannot distribute exponents over addition/subtraction: ( (a+b)^n \neq a^n + b^n ).
Key Terms & Definitions
- Base — The number or expression being multiplied.
- Exponent — Indicates how many times the base is used as a factor.
- Product Property of Exponents — ( a^m \times a^n = a^{m+n} ).
- Quotient Property of Exponents — ( a^m / a^n = a^{m-n} ).
- Negative Exponent — ( a^{-n} = 1/a^n ).
- Zero Exponent — ( a^0 = 1 ) (if ( a \neq 0 )).
- Power Property — ( (a^m)^n = a^{mn} ).
- Power of a Product — ( (ab)^n = a^n b^n ).
- Power of a Quotient — ( (a/b)^n = a^n / b^n ).
Action Items / Next Steps
- Answer conceptual questions about each property in your own words.
- Complete practice problems and rewrite answers using only positive exponents.