1. Basics of Exponents
Exponents are shorthand for repeated multiplication. They help us express very large or very small numbers in a compact way.
Base and Power
In \(a^n\), a is the base and n is the power (exponent). Example: \(2^5 = 2 \times 2 \times 2 \times 2 \times 2 = 32\).
Standard Form (Scientific Notation)
Very large or small numbers are written as \(m \times 10^n\), where \(1 \leq m < 10\). Examples:
- \(3,600,000 = 3.6 \times 10^6\)
- \(0.00042 = 4.2 \times 10^{-4}\)
Why 3.6 and not 36 or 0.36?
In standard form, the first number (mantissa) must always be between 1 and 10.
- \(36 \times 10^5\) ❌ (mantissa too large)
- \(0.36 \times 10^7\) ❌ (mantissa too small)
- \(3.6 \times 10^6\) ✅ (mantissa between 1 and 10, correct standard form)
Why It Matters
This rule ensures every number has a unique representation, makes comparison easier, and keeps calculations neat in science, engineering, and finance.