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 you multiply two things with the same base, you add the exponents. The reason: you are just combining two groups of repeated multiplications into one longer one.
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
Division is the reverse: subtract the exponents. The numerator and denominator share the same base, so they cancel from the bottom up.
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 anything except zero itself. The cleanest proof: aⁿ divided by aⁿ uses the quotient rule to give a⁰, but also equals 1 because anything divided by itself is 1. So a⁰ must equal 1.
Negative exponents flip to the denominator. 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
Sources & Further Reading
The explanations on this page draw on the following established sources. We link to primary and secondary sources so you can verify claims and go deeper on any topic.