Exponent Rules
There are six core exponent rules. Every rule follows logically from the definition of an exponent as repeated multiplication. Understanding the logic — not just memorizing the rules — makes simplifying expressions much faster and more reliable.
In this lesson
1 Quick Review: What Exponents Mean
aⁿ means a multiplied by itself n times: 2⁴ = 2×2×2×2 = 16. The base is a, the exponent is n. All six rules follow from this definition.
Every exponent rule can be proven by expanding into repeated multiplication. When a rule seems mysterious, expand it out. The logic always becomes clear.
2 Product Rule: aᵐ × aⁿ = aᵐ⁺ⁿ
When multiplying same-base terms, add the exponents.
x³ × y⁵ cannot be simplified — the bases are different. You cannot add the exponents unless the bases match.
3 Quotient Rule: aᵐ ÷ aⁿ = aᵐ⁻ⁿ
When dividing same-base terms, subtract the exponents.
4 Power Rule: (aᵐ)ⁿ = aᵐⁿ
When raising a power to a power, multiply the exponents. Also: (ab)ⁿ = aⁿbⁿ and (a/b)ⁿ = aⁿ/bⁿ.
5 Zero and Negative Exponents
Zero exponent: a⁰ = 1 (for any a ≠ 0). Proof: aⁿ ÷ aⁿ = a⁰ and anything divided by itself = 1, so a⁰ = 1.
Negative exponent: a⁻ⁿ = 1/aⁿ. Proof: a³ ÷ a⁵ = a⁻² and also = a³/a⁵ = 1/a², so a⁻² = 1/a².
6 Fractional Exponents
a^(1/n) = ⁿ√a (the nth root). a^(m/n) = (ⁿ√a)ᵐ = ⁿ√(aᵐ).
Try the Exponent Calculator
Calculate any base to any power — including negative and fractional exponents.
Practice Problems
📚 Further Reading & Resources
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