Overview
This lecture introduces the fundamental laws of exponents, including their definitions, properties, and how to apply them to simplify various algebraic expressions.
Basics of Exponents
- Any number raised to the first power equals itself: ( x^1 = x ).
- Any number raised to the zeroth power equals one: ( x^0 = 1 ).
- Exponents represent repeated multiplication of the base.
Negative Exponents
- A negative exponent means reciprocal: ( x^{-n} = 1/x^n ).
- Apply the same logic for any negative exponent; e.g., ( 2^{-3} = 1/(2 \times 2 \times 2) = 1/8 ).
Powers Raised to Powers (Power Rule)
- To raise a power to another power: ( (x^m)^n = x^{m \times n} ).
- Works for both positive and negative exponents.
Multiplying and Dividing with the Same Base
- When multiplying same bases, add exponents: ( x^m \times x^n = x^{m+n} ).
- When dividing same bases, subtract exponents: ( x^m / x^n = x^{m-n} ).
- If the exponent becomes negative during division, use the rule for negative exponents.
Distributing Exponents over Multiplication and Division
- Distribute exponent over multiplication: ( (xy)^m = x^m y^m ).
- Distribute exponent over division: ( (x/y)^n = x^n / y^n ).
- These rules also work in reverse (undistributing exponents).
Key Terms & Definitions
- Exponent — The number that tells how many times to multiply the base by itself.
- Base — The number being multiplied repeatedly in an exponential expression.
- Negative Exponent — Indicates reciprocal of the base raised to the positive exponent.
- Power Rule — Raising a power to another power multiplies the exponents.
- Product Rule — Multiplying with same bases adds exponents.
- Quotient Rule — Dividing with same bases subtracts exponents.
- Distributive Exponent Rule — Exponents can be applied to each part of a product or quotient inside parentheses.
Action Items / Next Steps
- Practice simplifying expressions using the laws of exponents.
- Review previous videos or materials on exponent basics if needed.
- Complete assigned exponent problems to reinforce understanding.