What are the Laws of Exponents?
Exponents (or powers) provide a shorthand way to represent repeated multiplication. These laws work for real numbers as well.
Laws of Exponents (For Real Numbers a, b > 0, and rational exponents):
| Law | Formula | Example |
|---|---|---|
| Quotient Law | aᵐ ÷ aⁿ = aᵐ⁻ⁿ (m > n) | 5⁶ ÷ 5² = 5⁴ |
| Power Law | (aᵐ)ⁿ = aᵐⁿ | (3²)⁴ = 3⁸ |
| Power of Product | (ab)ᵐ = aᵐ × bᵐ | (2×3)⁴ = 2⁴ × 3⁴ |
| Power of Quotient | (a/b)ᵐ = aᵐ/bᵐ | (4/5)³ = 4³/5³ |
| Zero Exponent | a⁰ = 1 (a ≠ 0) | 7⁰ = 1 |
| Negative Exponent | a⁻ᵐ = 1/aᵐ (a ≠ 0) | 2⁻³ = 1/2³ = 1/8 |
| Fractional Exponent | aᵐ/ⁿ = ⁿ√(aᵐ) = (ⁿ√a)ᵐ | 8²/³ = ∛(8²) = ∛64 = 4 |
Important Notes:
- These laws apply for all real numbers when the expressions are defined
- For negative bases, careful with even/odd roots
- 0⁰ is undefined