Overview
This lecture explains the essential laws of exponents, including rules for positive, zero, negative exponents, and methods for simplifying expressions using multiplication, division, and distribution of exponents.
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 a base.
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 Law
- When raising a power to another power, multiply the exponents: ( (x^m)^n = x^{mn} ).
- This law applies to both positive and negative exponents.
Multiplying Exponents with the Same Base
- When multiplying terms with the same base, add the exponents: ( x^m \times x^n = x^{m+n} ).
- Example: ( 2^3 \times 2^4 = 2^{3+4} = 2^7 ).
Dividing Exponents with the Same Base
- When dividing terms with the same base, subtract the exponents: ( x^m / x^n = x^{m-n} ).
- If the result is a negative exponent, rewrite as a reciprocal: ( x^{-2} = 1 / x^2 ).
Distributing Exponents (Different Bases)
- When multiplying bases and raising them to the same exponent: ( (xy)^m = x^m y^m ).
- When dividing bases and raising them to the same exponent: ( (x/y)^n = x^n / y^n ).
- These laws also work in reverse (undistribute exponents).
Key Terms & Definitions
- Exponent — the number indicating how many times the base is multiplied by itself.
- Base — the number being multiplied in an exponential expression.
- Negative Exponent — indicates reciprocal and repeated division, not multiplication.
- Power of a Power — multiplying exponents when raising a power to another power.
Action Items / Next Steps
- Practice exponent problems to reinforce understanding.
- Review previous videos or materials on exponents if needed.