Learning Objectives
- Understand exponents and express numbers using powers
- Apply laws of exponents
- Express numbers in standard form (scientific notation)
- Work with negative exponents
Key Concepts
Exponents
aⁿ means 'a' multiplied by itself 'n' times. Here, 'a' is the base and 'n' is the exponent.
Example: 2⁵ = 2 × 2 × 2 × 2 × 2 = 32
Laws of Exponents
- aᵐ × aⁿ = aᵐ⁺ⁿ (same base, multiply → add exponents)
- aᵐ ÷ aⁿ = aᵐ⁻ⁿ (same base, divide → subtract exponents)
- (aᵐ)ⁿ = aᵐˣⁿ (power of a power → multiply exponents)
- aᵐ × bᵐ = (ab)ᵐ (same exponent, different bases)
- a⁰ = 1 (any non-zero number to the power 0 is 1)
- a¹ = a (any number to the power 1 is itself)
Standard Form (Scientific Notation)
A number written as a × 10ⁿ where 1 ≤ a < 10.
Example: 384,000,000 = 3.84 × 10⁸
Summary
Exponents provide a shorthand for repeated multiplication. The laws of exponents simplify calculations involving powers. Standard form helps express very large or very small numbers concisely.
Important Terms
- Base
- The number being multiplied repeatedly
- Exponent
- The number indicating how many times the base is multiplied by itself
- Power
- The entire expression aⁿ, or the exponent itself
- Standard Form
- A way to express numbers as a × 10ⁿ where 1 ≤ a < 10
Quick Revision
- aᵐ × aⁿ = aᵐ⁺ⁿ
- aᵐ ÷ aⁿ = aᵐ⁻ⁿ
- (aᵐ)ⁿ = aᵐⁿ
- a⁰ = 1 (a ≠ 0)
- Large numbers can be written in standard form: a × 10ⁿ
Practice Tips
- Express large numbers like population, distance to stars in standard form
- Simplify expressions by applying laws of exponents step by step
- Remember: when multiplying same bases, ADD exponents; when dividing, SUBTRACT