Overview
This lecture covers the main laws of exponents, explaining their meaning, how to use them, and providing examples of simplification.
Basic Exponent Laws
- Any number raised to the first power equals itself: ( x^1 = x ).
- Any number raised to the zero power equals one: ( x^0 = 1 ).
Negative Exponents
- A negative exponent means repeated division: ( x^{-n} = 1 / x^n ).
- Example: ( 2^{-3} = 1 / (2^3) = 1/8 ).
Power of a Power
- When raising a power to another power, multiply the exponents: ( (x^m)^n = x^{m \cdot n} ).
- Works with negative exponents as well.
Multiplying and Dividing Same Base Exponents
- To multiply with the same base, add the exponents: ( x^m \times x^n = x^{m + n} ).
- To divide with the same base, subtract the exponents: ( x^m / x^n = x^{m - n} ).
- Negative exponents result if the bottom exponent is larger.
Distributing Exponents Over Multiplication/Division (Different Bases)
- Distribute exponent over multiplication: ( (xy)^m = x^m \times y^m ).
- Distribute exponent over division: ( (x/y)^n = x^n / y^n ).
- These rules also work in reverse to combine terms.
Key Terms & Definitions
- Exponent — the number indicating how many times to multiply the base by itself.
- Base — the number being multiplied by itself.
- Negative exponent — indicates division by the base raised to the positive exponent.
Action Items / Next Steps
- Practice simplifying expressions using each of the exponent laws.
- Review any prior lessons if basics of exponents are unclear.
- Complete assigned exponent practice problems.