Overview
This lecture explains the basic laws of exponents, including multiplication, division, and handling powers, products, quotients, zero, and negative exponents.
Product Law of Exponents
- When multiplying expressions with the same base, keep the base and add the exponents.
- Example: (2^3 \times 2^2 = 2^{3+2} = 2^5 = 32).
- Example: (x^5 \times x^4 = x^{5+4} = x^9).
Quotient Law of Exponents
- When dividing expressions with the same base, subtract the denominator's exponent from the numerator's.
- Example: (4^7 / 4^5 = 4^{7-5} = 4^2 = 16).
Power of a Power
- When an exponent is raised to another exponent, multiply the exponents.
- Example: ((2^3)^2 = 2^{3 \times 2} = 2^6 = 64).
Power of a Product
- Distribute the exponent to each factor inside the parentheses.
- Example: ((3y)^2 = 3^2 \cdot y^2 = 9y^2).
- Example: ((2m^3n^2)^4 = 2^4 m^{3 \times 4} n^{2 \times 4} = 16m^{12}n^8).
Power of a Quotient
- Distribute the exponent to both numerator and denominator.
- Example: ((x/4)^3 = x^3 / 4^3 = x^3/64).
- Example: ((2a^7/5b^3)^2 = 4a^{14}/25b^6).
Zero Exponent Rule
- Any number (except zero) raised to the zero power equals 1.
- Example: ((3y)^0 = 1), ((2m^3n^2)^0 = 1), (x^0 y^2 = y^2).
Negative Exponent Rule
- An expression with a negative exponent equals the reciprocal with a positive exponent.
- Example: (2^{-3} = 1/2^3 = 1/8).
- Example: (x^{-5}y^3 = y^3/x^5).
- (1/b^{-n} = b^n); for example, (1/3^{-2} = 3^2 = 9).
- (x^{-2}/y^{-3} = y^3/x^2).
Key Terms & Definitions
- Exponent — The number that indicates how many times to multiply the base by itself.
- Base — The number or variable that is multiplied by itself.
- Product Law — Add exponents when multiplying with the same base.
- Quotient Law — Subtract exponents when dividing with the same base.
- Power of a Power — Multiply exponents when raising an exponent to another exponent.
- Power of a Product — Distribute the exponent to each factor inside parentheses.
- Power of a Quotient — Distribute the exponent to both numerator and denominator.
- Zero Exponent — Any (nonzero) base raised to zero equals one.
- Negative Exponent — Indicates reciprocal with a positive exponent.
Action Items / Next Steps
- Practice solving exponent problems using all laws covered.
- Review any textbook examples on the laws of exponents.