Overview
This lecture introduces exponents and powers, covering key definitions, laws, handling negative and zero exponents, distinguishing between exponent and power, and writing large numbers in standard form.
Introduction to Exponents and Powers
- Exponents are a way to write large numbers in shorter form.
- In "8³", 8 is the base, 3 is the exponent; it means 8 × 8 × 8.
- Exponent shows how many times the base is multiplied by itself.
Exponent vs. Power
- Exponent is the number of times the base is used as a factor.
- Power refers to the whole expression (base and exponent together).
Reading Exponential Forms
- "x²" is read as "x square" and "x³" as "x cube".
- For exponents 4 or higher, use "raised to the power": e.g., "x⁴" is "x raised to the power 4".
Prime Factorization & Expressing Numbers as Powers
- Numbers can be expressed as products of their prime factors in exponential form.
- Example: 256 = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 = 2⁸.
Exponents of Negative Numbers
- Even exponents of negative numbers yield positive results.
- Odd exponents of negative numbers yield negative results.
Laws of Exponents
- Product Law: a^m × a^n = a^(m+n).
- Quotient Law: a^m ÷ a^n = a^(m−n) (for a ≠ 0).
- Power of a Power: (a^m)^n = a^(mn).
- Product to a Power: (a × b)^m = a^m × b^m.
- Quotient to a Power: (a/b)^m = a^m ÷ b^m.
- Any nonzero number to the power 0 is 1: a^0 = 1.
- Any number to the power 1 is itself: a^1 = a.
Simplifying Using Exponents
- Combine like bases by adding or subtracting exponents as per above laws.
- Express products and quotients in exponential forms using laws.
Standard Form
- Large numbers can be written as a × 10^n, where 1 ≤ a < 10.
- Example: 4301 = 4.301 × 10³.
Key Terms & Definitions
- Base — The number being multiplied.
- Exponent — The number that tells how many times the base is multiplied by itself.
- Power — The entire expression of base and exponent together.
- Standard Form — Writing numbers as a decimal between 1 and 10 multiplied by a power of 10.
Action Items / Next Steps
- Practice more questions on laws of exponents.
- Convert large numbers into standard form as homework.
- Review definitions and laws before the next lesson.