Exponent Calculator
Raise any base to any power, including negative and fractional exponents, find nth roots, convert scientific notation, and model exponential growth or decay — with exact results where they exist and every step shown.
Written by Suhaib Hassan. The laws of exponents are standard, and each worked example is re-derived by hand. Our methodology.
Whole-number powers use exact big integers. Other powers use double-precision floating point, with rational exponents recognised so that results such as 4^2.5 = 32 come out exact.
It explains what a fractional or negative exponent means, warns you when a result is complex, and shows the curve the answer lies on.
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Exponential curve
The curve y = bᵗ with your point marked.
Step-by-step working
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Laws of exponents
These hold whenever the expressions are defined.
| Rule | Identity | Example | Condition |
|---|---|---|---|
| Product | bᵐ · bⁿ = bᵐ⁺ⁿ | 4² · 4³ = 4⁵ = 1024 | Same base |
| Quotient | bᵐ ÷ bⁿ = bᵐ⁻ⁿ | 4⁵ ÷ 4² = 4³ = 64 | b ≠ 0 |
| Power of a power | (bᵐ)ⁿ = bᵐⁿ | (4²)³ = 4⁶ = 4096 | b > 0 for non-integer exponents |
| Negative exponent | b⁻ⁿ = 1 ÷ bⁿ | 4⁻² = 1/16 = 0.0625 | b ≠ 0 |
| Zero exponent | b⁰ = 1 | 4⁰ = 1 | b ≠ 0 (0⁰ is indeterminate) |
| Fractional exponent | b^(m/n) = (ⁿ√b)ᵐ | 64^(1/3) = 4 | b ≥ 0 for even n |
Worked examples
Addressable memory
How many bytes can a 32-bit and a 64-bit address reach?
2⁶⁴ = 18,446,744,073,709,551,616 (≈ 18.4 exabytes)
Continuous compounding
$1,000 at 5% compounded continuously for 10 years.
= $1,648.72
Radioactive decay
Carbon-14 has a half-life of 5,730 years. What fraction remains after 11,460 years?
What is an exponent?
An exponent says how many times to use a base as a factor: b³ = b × b × b. It extends to zero (b⁰ = 1), negative numbers (b⁻ⁿ = 1 ÷ bⁿ) and fractions (b^(1/n) is the nth root of b). Real exponents are defined through the exponential function: bˣ = eˣ ˡⁿ ᵇ for b > 0.
Scientific notation
A number is in scientific notation when written a × 10ⁿ with 1 ≤ |a| < 10. Engineering notation restricts n to multiples of 3 so it lines up with SI prefixes (kilo, mega, milli…). E-notation such as 6.02e23 means 6.02 × 10²³.
By Suhaib Hassan · Reviewed 28 September 2026 · How CalculatePilot verifies formulas →
Sources
- NIST Digital Library of Mathematical Functions, chapter 4, section 4.2 (definitions of the logarithm and exponential).
- IEEE 754-2019, Standard for Floating-Point Arithmetic — the double-precision format used for non-integer powers.
- ISO 80000-2:2019, Quantities and units — Part 2: Mathematics.
Frequently asked questions
How is a fractional power like 4^2.5 calculated?
Write the exponent as a fraction, 2.5 = 5/2. The denominator is the root and the numerator the power: 4^(5/2) = (√4)⁵ = 2⁵ = 32. The calculator detects such fractions automatically and shows the exact value when it exists.
What happens with a negative base and a fractional exponent?
If the exponent is a fraction with an odd denominator the result is real ((−8)^(1/3) = −2). With an even denominator, such as (−4)^(1/2), there is no real result; the principal complex value is 0 + 2i, which this page reports when it applies.
Why is 0⁰ undefined?
Two rules conflict: x⁰ = 1 for every non-zero x, but 0ˣ = 0 for every positive x. Neither wins, so 0⁰ is left indeterminate in analysis. Many programming languages and combinatorics texts define it as 1 by convention; this calculator reports it as indeterminate.
What precision limits apply to large results?
Whole-number powers of whole numbers are exact big integers. Other results use IEEE 754 double precision: about 15–17 significant digits, with overflow to ∞ above about 1.8 × 10³⁰⁸ and underflow to 0 below about 5 × 10⁻³²⁴.
What is the difference between exponential growth and a power function?
In a power function like x², the base varies and the exponent is fixed. In an exponential function like 2ˣ the base is fixed and the exponent varies — so it eventually outgrows every power function.
How do I find the doubling time?
For continuous growth N = N₀eᵏᵗ, the doubling time is ln 2 ÷ k. At k = 0.05 per year that is 0.693147 ÷ 0.05 ≈ 13.86 years. For decay, the half-life is ln 2 ÷ |k|.
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