Exponent Calculator
Enter values into any two of the fields below to solve for the third. Supports negative bases, decimal exponents, and Euler’s number e as a base.
Formula
Example: 23 = 8. Leave one field blank to solve for it.
What is an exponent?
Exponentiation is a mathematical operation, written as an, involving a base a and an exponent n. When n is a positive integer, exponentiation means multiplying the base by itself n times.
The calculator accepts negative bases but does not compute imaginary numbers. It does not accept fractions as input notation, but you can enter fractional exponents in decimal form (e.g. 0.5 for a square root).
Basic exponent laws and rules
Multiplication (same base)
When exponents share the same base are multiplied, add the exponents:
Example: 22 × 24 = 4 × 16 = 64, or 2(2+4) = 26 = 64.
Negative exponents
Reciprocate the base and use the positive exponent:
Example: 2(−3) = 1 ÷ 23 = 18.
Division (same base)
Subtract the exponents:
Example: 2224 = 416 = 14, or 2(2−4) = 2−2 = 14.
Power of a power
Multiply the exponents:
Example: (22)4 = 44 = 256, or 28 = 256.
Power of a product
Example: (2 × 4)2 = 82 = 64, or 22 × 42 = 4 × 16 = 64.
Power of a quotient
Example: (25)2 = 425.
Special exponents
Exponent of 1: a1 = a — the base stays the same.
Exponent of 0: a0 = 1 for any a ≠ 0. This follows from an × a0 = an, so a0 must equal 1. Note: 00 is debated; this calculator follows the convention 00 = 1 in most practical contexts.
Fractional exponents (roots)
A fractional exponent with numerator 1 is an nth root: a1/n = n√a.
For general fractions: ap/q = (q√a)p. Example: 35/7 ≈ 2.19.
Enter fractional exponents as decimals (e.g. 0.5 for square root, 0.333… for cube root approximation).
Negative bases
- Negative base with even positive integer exponent → positive result (e.g. (−2)4 = 16).
- Negative base with odd positive integer exponent → negative result (e.g. (−2)3 = −8).
- Fractional exponent with negative base → imaginary number (not supported; calculator shows an error).
How to use this exponent calculator
- Enter any two of base, exponent, and result.
- Leave the third field empty — the calculator solves for it.
- Click use e as base to set the base to Euler’s number (≈ 2.71828).
- Click Calculate to see the answer and steps.
Examples: Base 2, exponent 10 → result 1024. Base 9, result 3 → exponent 0.5 (square root).
Real-world uses of exponents
- Compound interest: A = P(1 + r)t — see our compound interest calculator.
- Scientific notation: 6 × 1023 uses powers of ten.
- Computer science: Memory sizes (210 bytes = 1 KB in binary convention).
- Population growth & decay: Exponential models in biology and physics.