Overview
This lecture covered the fundamental properties of exponents and demonstrated how to simplify various algebraic expressions involving exponents.
Properties of Exponents
- When multiplying terms with the same base, add their exponents: (x^4 \times x^5 = x^9).
- When dividing terms with the same base, subtract exponents: (x^7 \div x^3 = x^4).
- Raising a power to another power means multiply exponents: ((x^3)^4 = x^{12}).
- Any nonzero number raised to the 0 power equals 1: (x^0 = 1).
- A negative exponent means reciprocal: (x^{-3} = 1/x^3); (1/x^4 = x^{-4}).
Simplifying Expressions with Exponents
- Multiply numbers and add exponents: (5x^3 \times 4x^7 = 20x^{10}).
- When multiplying powers: (7x^6 \times 5x^4 = 35x^{10}).
- Distribute exponents: ((3x^2)^3 = 27x^6).
- For multiple variables: ((4x^3 y^2) \times (7x^4 y^3) = 28x^7 y^5).
- Negative exponents in denominator move variable to numerator as positive exponent: (y^{-2} = 1/y^2).
- Subtract exponents for division: (x^7 / x^{12} = x^{-5} = 1/x^5).
Handling Parentheses and Signs
- ((-3)^2 = 9), but (-3^2 = -9); parentheses matter.
- When raising a product to a power, apply exponent to all factors: ((-2x^3 y^4)^2 = 4x^6 y^8).
- Zero exponent applies only to the variable next to it: (-4x^0 = -4).
Complex Fraction Example
- Simplify coefficients and exponents separately: (24x^7y^3 / 8x^4y^{-2} = 3x^3y^5).
- When distributing an exponent over a quotient: ((5x^7/9y^2)^2 = 25x^{14}/81y^4).
Key Terms & Definitions
- Exponent — Number indicating how many times to multiply the base by itself.
- Negative exponent — Denotes reciprocal of base raised to the positive exponent.
- Zero exponent — Any nonzero base raised to 0 is 1.
- Base — The number or variable being multiplied by itself.
Action Items / Next Steps
- Practice simplifying similar exponent expressions.
- Review the rules for adding, subtracting, and multiplying exponents.
- Work on assigned homework problems involving exponent simplification.