Overview
This lecture reviews the laws of exponents, including definitions, key rules, and example problems, to help students master exponent manipulations.
Exponents & Powers: Key Concepts
- An exponent shows how many times a base number is multiplied by itself.
- Example: In 5³, 5 is the base, 3 is the exponent, and 125 is the power (result).
- 5³ is not 5 × 3, but 5 × 5 × 5.
Powers of Numbers
- Memorizing powers of small prime numbers (like 2, 3, 5) and 10 is useful for calculations.
- Recognizing numbers as powers of other numbers helps simplify calculations.
Changing Bases
- To use exponent laws, rewrite all terms so they have the same base.
- Change numbers like 64 to 2⁶ to match the base of other terms.
Laws of Exponents
- Product of Powers: When multiplying with the same base, add the exponents: aᵐ × aⁿ = a^(m+n).
- Quotient of Powers: When dividing with the same base, subtract the exponents: aᵐ / aⁿ = a^(m−n).
- Power of a Power: When raising a power to another exponent, multiply the exponents: (aᵐ)ⁿ = a^(m×n).
- Power of a Product: Distribute the exponent: (ab)ᵐ = aᵐbᵐ.
- Power of a Quotient: Distribute the exponent: (a/b)ᵐ = aᵐ / bᵐ.
Zero, One, Negative, and Fractional Exponents
- Any nonzero base to the power of 0 equals 1: a⁰ = 1.
- Any base to the power of 1 equals itself: a¹ = a.
- Negative exponents mean reciprocal: a^(−n) = 1 / aⁿ.
- Fractional exponents represent roots: a^(m/n) = n-th root of aᵐ.
Examples & Applications
- Simplify using exponent rules: combine exponents, rewrite negative exponents as reciprocals, and convert fractional exponents to roots.
- In equations like aᵐ = aⁿ, the exponents are equal: m = n.
- Check answers by substituting values back into the original equation.
Key Terms & Definitions
- Base — The number being multiplied repeatedly.
- Exponent — The number indicating how many times the base is used as a factor.
- Power — The result of raising a base to an exponent.
- Product of Powers — Law stating to add exponents when multiplying with the same base.
- Quotient of Powers — Law stating to subtract exponents when dividing with the same base.
Action Items / Next Steps
- Practice exercises applying all exponent rules.
- Memorize common small powers and their base-exponent pairs (e.g., 2³ = 8, 3⁴ = 81).
- Review and simplify expressions by expressing terms with the same base before applying laws.