Laws of Exponents
Complete reference for exponent laws: product, quotient, power of a power, zero rule, and negative exponents.
With worked examples and common mistakes.
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
Common Mistakes
- The product rule applies only when the bases are the same — x³ × y³ cannot use the product rule; instead factor: x³y³ = (xy)³
- (-2)² = 4, but −2² = −4 — the exponent applies to 2 only, not the minus sign, unless parentheses wrap the whole base: (−2)²
- 0⁰ is indeterminate — most calculators return 1 (the combinatorics convention), but in limits it may differ; treat it with caution in advanced work
Key Notes
- Core rules: aᵐ × aⁿ = aᵐ⁺ⁿ; aᵐ / aⁿ = aᵐ⁻ⁿ; (aᵐ)ⁿ = aᵐⁿ; (ab)ⁿ = aⁿbⁿ; (a/b)ⁿ = aⁿ/bⁿ. All five rules must be applied to the same base — you cannot combine aᵐ × bⁿ into a single power unless a = b.
- Zero and negative exponents: a⁰ = 1 for any a ≠ 0 (follows from the division rule: aⁿ/aⁿ = aⁿ⁻ⁿ = a⁰ = 1). a⁻ⁿ = 1/aⁿ — a negative exponent means "take the reciprocal," not "make negative." 2⁻³ = 1/8, not −8.
- Fractional exponents are roots: a^(1/n) = ⁿ√a; a^(m/n) = (ⁿ√a)ᵐ = ⁿ√(aᵐ). The denominator is the root index; the numerator is the power. 8^(2/3) = (∛8)² = 2² = 4.
- Scientific notation uses exponent rules: (3 × 10⁴) × (4 × 10³) = 12 × 10⁷ = 1.2 × 10⁸. Multiplying in scientific notation means adding exponents; dividing means subtracting them.
- Applications: Laws of exponents are used in compound interest calculations, scientific notation manipulation, simplifying algebraic expressions, working with logarithms (log(aⁿ) = n log a), and evaluating derivatives of power functions (d/dx xⁿ = nxⁿ⁻¹).