Laws of Exponents
Complete reference for all exponent laws including product, quotient, power, and zero rules with examples.
The Formulas
Product Rule: aᵐ × aⁿ = aᵐ⁺ⁿ
Quotient Rule: aᵐ ÷ aⁿ = aᵐ⁻ⁿ
Power Rule: (aᵐ)ⁿ = aᵐⁿ
Zero Exponent: a⁰ = 1 (a ≠ 0)
Negative Exponent: a⁻ⁿ = 1/aⁿ
Fractional Exponent: a^(m/n) = ⁿ√(aᵐ)
Quotient Rule: aᵐ ÷ aⁿ = aᵐ⁻ⁿ
Power Rule: (aᵐ)ⁿ = aᵐⁿ
Zero Exponent: a⁰ = 1 (a ≠ 0)
Negative Exponent: a⁻ⁿ = 1/aⁿ
Fractional Exponent: a^(m/n) = ⁿ√(aᵐ)
The laws of exponents are fundamental rules that simplify expressions with powers. Mastering them is essential for algebra, calculus, and scientific notation.
Variables
| Symbol | Meaning |
|---|---|
| a | The base (any nonzero real number) |
| m, n | The exponents (integers or fractions) |
Example 1
Simplify x³ × x⁵
Product rule: add exponents when bases are the same
= x³⁺⁵ = x⁸
Example 2
Simplify (2y²)³
Power of a product: (ab)ⁿ = aⁿ × bⁿ
= 2³ × (y²)³ = 8 × y⁶
= 8y⁶
Example 3
Simplify 5⁻² × 5⁴
Product rule: 5⁻²⁺⁴ = 5²
= 25
When to Use Them
Use the laws of exponents when:
- Simplifying algebraic expressions with powers
- Working with scientific notation (e.g., 3 × 10⁸)
- Solving exponential equations
- Differentiating or integrating power functions in calculus