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Laws of Exponents Calculator

Combine same-base powers using the product, quotient, and power-of-a-power rules to get a simplified exponent and its numeric value.
The common base shared by both powers

Operation

Result exponent
5

Combined exponent after applying the law

Product rule
2^{5} = 32
Simplified power
2^5
Numeric value
32
Step by step
  1. 1

    Exponent m

    3
  2. 2

    Exponent n

    2
  3. 3

    Combined exponent (product rule bᵐ × bⁿ = bᵐ⁺ⁿ)

    m + n = 3 + 2 = 5
Results are estimates for general information only and are not professional advice — always verify important results independently before relying on them. Read the full disclaimer.
Quick answer

How does this calculator work?

To combine same-base powers, keep the base and operate on the exponents: multiplying adds them, dividing subtracts them, and raising a power to a power multiplies them. Enter the base, both exponents, and an operation to get the simplified exponent plus the exact numeric value of the result.

Formula
bᵐ × bⁿ = bᵐ⁺ⁿ ; bᵐ ÷ bⁿ = bᵐ⁻ⁿ ; (bᵐ)ⁿ = bᵐ×ⁿ
How this is calculated

Enter the common base b and the two exponents m and n, then choose an operation. All three exponent laws require the powers to share the same base b, which is why only one base field is provided.

Multiplying same-base powers adds the exponents (bᵐ × bⁿ = bᵐ⁺ⁿ), dividing subtracts them (bᵐ ÷ bⁿ = bᵐ⁻ⁿ), and raising a power to a power multiplies them ((bᵐ)ⁿ = bᵐ×ⁿ). The calculator computes the resulting exponent and then evaluates the numeric value b raised to that exponent.

Exponents may be negative or fractional: a negative exponent represents a reciprocal and a fractional exponent represents a root. Edge cases such as a zero or negative base combined with a fractional exponent can produce a non-real or undefined result, in which case no value is shown.

Frequently asked questions

The product, quotient, and power-of-a-power rules only apply when both powers share the same base. With different bases you cannot simply add or subtract the exponents.

Yes. Negative exponents give reciprocals (b⁻ⁿ = 1/bⁿ) and fractional exponents give roots (b^(1/2) = √b). The same addition, subtraction, and multiplication rules still apply.

The quotient rule yields b⁰, and any non-zero base raised to the zero power equals 1.

Also known as

laws of exponents
exponent rules
product rule
quotient rule
power of a power
simplify powers
exponent laws
rules of exponents

APA

TG we-Calculate Editorial Team. (2026). Laws of Exponents Calculator [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/laws-of-exponents-calculator

Chicago

TG we-Calculate Editorial Team. "Laws of Exponents Calculator." TG we-Calculate. 2026. https://we-calculate.com/calculator/laws-of-exponents-calculator.

IEEE

TG we-Calculate Editorial Team, "Laws of Exponents Calculator," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/laws-of-exponents-calculator

BibTeX

@misc{wecalculate_laws_of_exponents_calculator, title = {Laws of Exponents Calculator}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/laws-of-exponents-calculator}}, year = {2026}, note = {TG we-Calculate} }

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