Mathematics: Exponents (Power)

Mathematical Formulas on Powers of Numbers

1. Basic Definition of Power

aⁿ = a × a × a × ... (n times)

Example: 2⁴ = 2 × 2 × 2 × 2 = 16

2. Laws of Exponents

Product Rule (Same Base)

a^m × aⁿ = a^m+n

Example: 3² × 3⁴ = 3⁶

Quotient Rule (Same Base)

a^m / aⁿ = a^m−n (a ≠ 0)

Example: 5⁷ / 5³ = 5⁴

Power of a Power

(a^m)ⁿ = a^mn

Example: (2³)⁴ = 2¹²

Power of a Product

(ab)ⁿ = aⁿbⁿ

Example: (3 × 4)² = 3² × 4²

Power of a Quotient

(a / b)ⁿ = aⁿ / bⁿ (b ≠ 0)

3. Special Exponent Rules

Zero Power Rule

a⁰ = 1 (a ≠ 0)

Negative Exponent Rule

a⁻ⁿ = 1 / aⁿ (a ≠ 0)

Example: 2⁻³ = 1 / 2³ = 1/8

Fractional Exponents

a^1/n = √[n]a

a^m/n = √[n](a^m)

Example: 8^1/3 = ∛8 = 2

4. Important Identities

1ⁿ = 1

0ⁿ = 0 (n > 0)