Understanding Exponents
An exponent tells you how many times to multiply a base number by itself. The expression xⁿ is read as "x raised to the power n" or "x to the nth power."
xⁿ = x × x × x × ... (n times)
3⁴ = 3 × 3 × 3 × 3 = 81
3⁴ = 3 × 3 × 3 × 3 = 81
Common Exponent Examples
2⁸
256
3⁵
243
10³
1,000
5⁴
625
Exponent Rules — Complete Reference
Understanding exponent rules lets you simplify complex expressions without a calculator. These rules are fundamental in algebra, calculus, and all higher mathematics.
| Rule | Formula | Example |
|---|---|---|
| Product Rule | aᵐ × aⁿ = aᵐ⁺ⁿ | 2³ × 2⁴ = 2⁷ = 128 |
| Quotient Rule | aᵐ ÷ aⁿ = aᵐ⁻ⁿ | 3⁵ ÷ 3² = 3³ = 27 |
| Power of Power | (aᵐ)ⁿ = aᵐⁿ | (2²)³ = 2⁶ = 64 |
| Zero Exponent | a⁰ = 1 | 5⁰ = 1, 100⁰ = 1 |
| Negative Exponent | a⁻ⁿ = 1/aⁿ | 2⁻³ = 1/8 = 0.125 |
| Fractional Exponent | a^(1/n) = ⁿ√a | 9^(1/2) = √9 = 3 |
| Fractional General | a^(m/n) = ⁿ√(aᵐ) | 8^(2/3) = ∛(64) = 4 |
Exponent Powers of 2 — Essential for Computing
Powers of 2 are the foundation of binary arithmetic and all digital computing. Every byte, kilobyte, megabyte, and gigabyte is a power of 2.
| Exponent | Value | Computing term |
|---|---|---|
| 2⁰ | 1 | 1 bit off = 0 |
| 2¹ | 2 | — |
| 2⁸ | 256 | 8 bits = 1 byte |
| 2¹⁰ | 1,024 | ≈ 1 Kilobyte |
| 2²⁰ | 1,048,576 | ≈ 1 Megabyte |
| 2³⁰ | 1,073,741,824 | ≈ 1 Gigabyte |
Scientific Notation and Exponents
Scientific notation expresses very large or small numbers as a number between 1 and 10 multiplied by a power of 10:
6,500,000 = 6.5 × 10⁶
0.000042 = 4.2 × 10⁻⁵
Speed of light = 3 × 10⁸ m/s
0.000042 = 4.2 × 10⁻⁵
Speed of light = 3 × 10⁸ m/s
Real-World Applications
- Compound interest: A = P(1+r)ⁿ — the exponent n is the number of compounding periods
- Population growth: P = P₀ × eʳᵗ — exponential growth model
- Computing: Memory sizes, CPU speeds, and data storage all use powers of 2
- Physics: Inverse square law (F ∝ 1/r²), radioactive decay (N = N₀ × e⁻λt)
- Chemistry: pH scale — pH = −log₁₀[H⁺], each unit is a 10× change