Exponent Calculator: Powers & Fractions

Exponent Calculator: Powers, Negative & Fractional Exponents

Free exponent calculator: calculate powers, negative exponents, and fractional exponents. Power of a power rules, how to find the base of an exponent, fractional exponent solver. Laws of exponents; solve for base, exponent, or result. Trusted by students and educators. No sign-up, all calculations run locally.

How the Math Works

Exponentiation raises a base to a power. The calculator solves for any one of the three components—base, exponent, or result—given the other two. The core definitions:

a^n = \underbrace{a \times a \times \cdots \times a}_{n \text{ times}}

For negative and fractional exponents:

a^{-n} = \frac{1}{a^n}, \quad a^{p/q} = \sqrt[q]{a^p}

To solve for a missing value, the calculator rearranges:

\text{base} = \text{result}^{1/n}, \quad \text{exponent} = \frac{\ln(\text{result})}{\ln(\text{base})}

a (Base):

The number being multiplied repeatedly
n (Exponent):

How many times the base is used as a factor (can be negative or fractional)
Result:

The value of aⁿ

Worked example—solve for base: exponent = 3, result = 27.

base = 27^1/3 = ∛27 = 3

Worked example—negative exponent: 2⁻³ = 1 / 2³ = 1/8 = 0.125.

Edge cases: 0⁰ is conventionally defined as 1. Negative bases with non-integer exponents produce complex numbers and are outside this calculator’s real-number scope.

How to Use This Calculator

Decide which value you need and leave that field blank. To compute a power, enter the base and exponent and leave Result empty—for example, base 2, exponent 10 gives 1 024. To find the base, enter the exponent and result (e.g., exponent 3, result 64 returns base 4). To find the exponent, enter the base and result (e.g., base 5, result 125 returns exponent 3). Use the “Use e as base” button to set the base to Euler’s number (≈ 2.71828) for natural exponentials. For small positive integer exponents, the calculator also shows the multiplication expansion (e.g., 2 × 2 × 2 × 2 × 2). When a negative or fractional exponent applies, a learning insight displays the relevant law.

What This Calculator Does & Who It's For

Calculator Purpose & Ideal Users

This exponent calculator solves for base, exponent, or result: enter any two, leave one blank.

What You'll Get:

Solve for result: aⁿ from base and exponent (including negative and decimal exponents). Solve for base: a from result and exponent, how to find the base of an exponent when you know the power and result. Solve for exponent: n from base and result (e.g. what exponent gives this power?). Use e as base: Button sets base to e (≈ 2.71828) for natural exponentials. Learning insight: Negative exponent law or natural exponential note when relevant. Expansion: For small positive integer exponents, shows the repeated multiplication (e.g. 2×2×2×2×2).
Ideal Users:

Students & algebra: Laws of exponents, calculate negative exponents, fractional exponent solver, power of a power rules. Teachers: Teaching module with laws and worked examples. Anyone: Quick power and nth root (base from result and exponent) calculations.
Scope & Limits:

Real numbers only; no imaginary results (e.g. negative base with non-integer exponent is not computed). Accepts positive/negative integers and decimals; fractional exponents entered as decimals. All calculations run in your browser; no data is sent to servers.

What Is an Exponent? Exponentiation Definition

An exponent (or power) is the number of times a base is multiplied by itself. We write

a^n

(a to the power n), where a is the base and n is the exponent. By definition,

a^n = a \times a \times \cdots \times a

(n times) for positive integers n. When the exponent is 1, a¹ = a. For n = 0, a⁰ = 1 (when a ≠ 0). For negative exponents,

a^{-n} = \frac{1}{a^n}

For fractional exponents,

a^{p/q}

is the qth root of aᵖ (e.g. a^0.5 = √a). This exponent calculator computes aⁿ for real bases and exponents

Laws of Exponents: Product, Quotient, Power of a Power

Product rule: When multiplying same bases, add exponents:

a^n \times a^m = a^{n+m}

Quotient rule: When dividing same bases, subtract exponents:

\frac{a^m}{a^n} = a^{m-n}

Power of a power: When raising a power to a power, multiply exponents:

(a^m)^n = a^{m \times n}

Negative exponent:

a^{-n} = \frac{1}{a^n}

These power of a power rules and laws of indices are shown in the calculator’s Laws of exponents panel and in the Learning insight when they apply to your calculation

Power of a Product and Power of a Quotient

Power of a product:

(ab)^n = a^n b^n

, distribute the exponent to each factor. Power of a quotient:

(a/b)^n = a^n / b^n

, apply the exponent to numerator and denominator. These follow from the definition of exponentiation and the product rule. Use this exponent calculator to compute single powers

Exponent Calculator FAQ

What happens when the exponent is 1?

When the exponent is 1, the result is the base itself: a¹ = a. For example, 5¹ = 5 and (−3)¹ = −3. This follows from the definition of exponentiation as repeated multiplication, multiplying the base once gives the base.

How do you solve negative exponents?

A negative exponent means the reciprocal of the base raised to the positive exponent:

a^{-n} = \frac{1}{a^n}

For example, 2⁻³ = 1/8. Enter a negative exponent in this calculator to see the result and the Negative Exponent Law in the Learning insight

Can an exponent be a decimal?

Yes. An exponent can be a decimal (fractional exponent). For example, 4^0.5 = √4 = 2, and 8^(1/3) = ∛8 = 2. The calculator accepts decimal exponents; for a negative base with a non-integer exponent the result would be complex, so this tool restricts to real results (e.g. positive base, or negative base with integer exponent).

What is the base e in exponentiation?

e (Euler’s number, ≈ 2.71828) is the natural exponential base. eⁿ appears in growth, decay, and logarithms. Use the “Use e as base” button in this calculator to set the base to e and compute natural exponentials.

How do I find the base of an exponent?

If you know the result and the exponent, the base is a = result^(1/n). Leave the Base field blank and enter the exponent and result; the calculator solves for the base. For a positive result, the base is the nth root of the result.

Exponent Calculator

Base, exponent & result

Expansion

Laws of exponents

Formulas used

When and How to Use the Exponent Calculator

Workflow Tips

Solving for Result

Solving for Base

Solving for Exponent

Negative and Fractional Exponents