Exponent Rules Simplifier Calculator
Apply the product, quotient and power-of-a-power rules to two exponents on the same base and see the simplified expression.
How it works
bᵃ · bᵇ = bᵃ⁺ᵇ · bᵃ ÷ bᵇ = bᵃ⁻ᵇ · (bᵃ)ᵇ = bᵃᵇ
The exponent rules turn awkward products, quotients and nested powers into a single power of the same base. When you multiply powers of the same base you add the exponents; when you divide you subtract them; and raising a power to another power multiplies the exponents. A negative resulting exponent means a reciprocal, so it can be rewritten with a positive exponent underneath 1.
Worked example
For x⁵ · x², the product rule adds the exponents: 5 + 2 = 7, so the result is x⁷. If instead you divided, x⁵ ÷ x² = x^(5 − 2) = x³.
Sources: Exponentiation — Wikipedia
Frequently asked questions
Do these rules only work for the same base?
The product and quotient rules require the same base — you cannot combine x³ and y² into one power. The power-of-a-power rule works on whatever base is inside the parentheses.
What does a negative exponent mean?
A negative exponent is a reciprocal: b⁻ⁿ = 1/bⁿ. So x² ÷ x⁵ = x⁻³ = 1/x³. The calculator shows this positive-exponent form whenever the result is negative.
What is anything to the power of zero?
Any non-zero base raised to the power 0 equals 1. This falls out of the quotient rule: bⁿ ÷ bⁿ = b⁰, and any non-zero number divided by itself is 1.
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This calculator is for educational purposes. Double-check important results before acting on them.