Exponent Calculator
Raise any number to any power — whole, zero, negative or decimal exponents, such as 2⁻³ = 0.125 or 1.05^30 for 30 years of 5% growth — with the exponent rule explained. Free, runs in your browser.
Exponent and Power Calculator
How to Use the Exponent Calculator
Enter the base
Type the number you want to raise to a power — the base. It can be a whole number, a decimal, or negative.
Enter the exponent
Type the power. It can be positive, negative (for reciprocals), or a decimal (for roots). The box reads decimals only — a slash is not understood as a fraction — so write ⅓ as 0.333333333333, with enough digits that 27^0.333333333333 rounds to 3; for an exact root in radical notation use the Nth Root Calculator instead.
Read the result
The answer appears instantly, rounded to twelve significant figures, with the value shown in the familiar base-and-superscript notation. A result too large for double-precision arithmetic (above about 1.8 × 10³⁰⁸, e.g. 2¹⁰²⁴) reads Infinity.
Understand the rule
A short note explains which exponent rule applies — repeated multiplication, reciprocal, or root — so you learn the why, not just the answer.
Exponents, From Whole to Fractional
One Idea, Stretched
An exponent starts as the simplest shorthand in mathematics: 2³ just means 2 × 2 × 2 = 8, "multiply the base by itself this many times". But the real beauty of exponents is how that one idea stretches consistently to cases where "multiply yourself three times" stops making literal sense. What is 2⁰? The rules force it to be 1, because dividing 2³ by 2 each time — 8, 4, 2, 1 — lands you there. What is a negative exponent? 2⁻¹ continues the same halving pattern past 1 to give ½, so a negative exponent means "reciprocal": x⁻ⁿ = 1 ÷ xⁿ. And what is a fractional exponent? 27^(1/3) is the cube root of 27, which is 3, because raising it to the third power must give back 27. This calculator handles all of these — whole, zero, negative, and decimal exponents — and tells you which rule is doing the work, so the consistency behind the notation becomes visible rather than mysterious. One convention worth knowing: it treats 0⁰ as 1, the choice Knuth argued for in 1992 and the one built into the IEEE 754 and ECMAScript pow functions it runs on.
That consistency is not a curiosity; it is what makes exponents one of the most powerful tools in science and finance. Compound interest, population growth, radioactive decay, sound and earthquake scales, computer storage sizes — all are described by exponents, because all involve a quantity multiplying by a fixed factor over and over. The same notation that says "2 cubed" also says "grows by 5% for 30 years" or "halves every 8 days". Fractional exponents, in particular, are the bridge between powers and roots: writing a square root as x^(1/2) or a cube root as x^(1/3) lets you manipulate roots with the same simple algebra you use for whole-number powers, which is why scientific and engineering calculators treat them as one operation.
"A negative exponent is a reciprocal; a fractional exponent is a root. Exponents are one idea stretched to stay consistent — and that consistency is their power."
A Note on Negatives
There is one case where the real numbers genuinely run out: a negative base raised to a non-whole power. Asking for (-8)^(1/2) — the square root of −8 — has no real answer, because no real number squared gives a negative, and this calculator says so rather than returning a confusing value. (The answer does exist in the complex numbers, which the site's Complex Number Calculator handles.) Odd roots need care. The real cube root of −27 is −3, because a negative times a negative times a negative is negative — but this calculator does not return it. It follows the general definition of a power, xʸ = e^(y·ln x), as standardised for the pow function in IEEE 754 and ECMA-262: for a negative base and any exponent that is not a whole number there is no real value, so (-27)^0.333333333333 reports "no real value" here. (A decimal such as 0.333333333333 is in any case not exactly ⅓, so no odd-root rule could safely apply to it.) For the real odd root of a negative number, use the Nth Root Calculator, which takes the root index as a whole number and returns −3 for the cube root of −27. Negative bases with whole-number exponents work as expected: (-2)³ = −8. Whether you are doing homework on indices, working with scientific notation, or just settling what 2¹⁰ is, this calculator gives the answer and the reason behind it.
Why 2⁰ = 1 and 2⁻¹ = ½, and why (−8)^(1/3) gets no real answer here
A whole-number exponent means repeated multiplication: 2³ = 2×2×2.
Any non-zero number to the power 0 is 1.
A negative exponent is a reciprocal: 2⁻¹ = ½.
A fractional exponent is a root: 27^(1/3) is the cube root of 27 = 3 — enter it here as the decimal 0.333333333333.
To multiply powers of the same base, add the exponents.
A negative base with a non-whole power has no real value under the general rule xʸ = e^(y·ln x), because ln of a negative number is complex — the IEEE 754 and ECMA-262 pow convention this tool follows.
The real cube root of −27 is −3, but that is the odd-root convention: this tool reports no real value for (−27)^0.333333333333, and the Nth Root Calculator returns the −3.
Compound interest, decay, and growth are all described by exponents.
0⁰ is treated as 1 here — the convention Knuth defended in 1992 and the value IEEE 754 and ECMAScript pow() return.
Whole-number results above 2⁵³ ≈ 9.0 × 10¹⁵ are no longer exact in double-precision arithmetic, and above about 1.8 × 10³⁰⁸ the result reads Infinity.
Frequently Asked Questions
- A negative exponent means the reciprocal: x⁻ⁿ equals 1 ÷ xⁿ. So 2⁻¹ is ½, and 2⁻³ is 1/8. The calculator computes this directly and explains the reciprocal rule in the working below the answer.
- A fractional exponent is a root. x^(1/2) is the square root of x, x^(1/3) is the cube root, and so on. For example 27^(1/3) = 3 because 3 × 3 × 3 = 27. Enter the fraction as a decimal with plenty of digits — 27^0.3333 gives 2.99967, while 27^0.333333333333 rounds to 3 at twelve significant figures — or use the Nth Root Calculator for exact root notation.
- Because of the pattern of dividing by the base each time the exponent drops by one: 2³ = 8, 2² = 4, 2¹ = 2, 2⁰ = 1. Keeping that consistent rule forces the value 1 for any non-zero base. For 0⁰ the pattern argument breaks down, and mathematicians choose: this calculator returns 1, following Knuth's 1992 argument (it keeps the binomial theorem and power series valid) and the IEEE 754 / ECMAScript pow convention. The calculator notes the rule when you enter an exponent of 0.
- For (−8)^0.5 the reason is plain: no real number squared is negative, so the square root of −8 has no real value (it is a complex number, which the Complex Number Calculator can show). For (−27)^(1/3) the real cube root −3 does exist, but this calculator still reports no real value, because it evaluates powers by the general rule xʸ = e^(y·ln x) — the definition IEEE 754 and ECMA-262 give the pow function — and ln of a negative number is not real. A decimal like 0.333333333333 is also not exactly ⅓, so the tool cannot know you meant an odd root. Use the Nth Root Calculator for the real odd root of a negative number; negative bases with whole-number exponents, such as (−2)³ = −8, compute normally here.
- Yes — both the base and the exponent can be decimals. This is exactly what you need for growth and decay calculations, like 1.05^30 for thirty years of 5% growth, or roots written as decimal powers.
- The calculation runs in IEEE 754 double precision — about 15 to 17 significant decimal digits — and the display is rounded to twelve significant figures, which hides the usual binary rounding (0.1² is 0.010000000000000002 internally and displays as 0.01). Whole-number results are exact only while they stay below 2⁵³ ≈ 9.0 × 10¹⁵; 3⁴⁰, for example, is exactly 12,157,665,459,056,928,801 but displays as 12,157,665,459,100,000,000 once the double-precision value is rounded to twelve figures. Beyond about 1.8 × 10³⁰⁸ the result overflows to Infinity.
- When multiplying powers of the same base, you add the exponents: xᵃ × xᵇ = xᵃ⁺ᵇ. When dividing, you subtract them. These index laws are why fractional and negative exponents behave so consistently, and they are the foundation for logarithms.
- Yes. The calculation runs in your browser — nothing is uploaded, stored, or logged — and it works offline once loaded.
- 0⁰ returns 1, the convention set out by Knuth (1992) and built into the IEEE 754 and ECMAScript pow functions. 0⁻¹ is 1 ÷ 0, which has no finite value, so the result reads Infinity. 2¹⁰²⁴ exceeds the largest double-precision number (about 1.8 × 10³⁰⁸), so it also reads Infinity, while 2¹⁰²³ ≈ 8.99 × 10³⁰⁷ still computes. Infinity here means "too large to represent", not a mathematical claim about the value.
- They are two views of the same maths. This tool raises a base to a power; the Nth Root calculator takes roots and writes them in radical notation (√, ∛). A fractional exponent here, like ^(1/3), is exactly a cube root there. Use whichever notation matches your problem.
Method & sources
How it computes
xʸ evaluated in IEEE 754 double precision with the ECMA-262 pow semantics: an exponent of 0 gives 1 for every base including 0⁰; a negative exponent is the reciprocal; a negative base with a non-integer exponent has no real value and is reported as such (the real odd root is a different convention, not applied here); 0 to a negative power and any result beyond about 1.8 × 10³⁰⁸ read Infinity. Display rounded to 12 significant figures.
What this tool implements
- Special cases follow ECMA-262 §6.1.6.1.3 Number::exponentiate: exponent ±0 → 1; base < −0 with a non-integral exponent → NaN, which the page reports as 'no real value'; overflow → Infinity.
- IEEE 754-2019 pow(x, y): pow(±0, ±0) = 1 and pow(x, y) is NaN for finite x < 0 with finite non-integer y — the same convention, so the tool agrees with every conforming library and calculator.
- 0⁰ = 1 by the convention argued in Knuth (1992): it keeps the binomial theorem and power-series identities valid at x = 0.
- Double precision carries about 15–17 significant decimal digits; whole-number results are exact only below 2⁵³ ≈ 9.007 × 10¹⁵, and the largest finite value is about 1.797 × 10³⁰⁸ (2¹⁰²⁴ overflows, 2¹⁰²³ does not).
Sources
- ECMA-262, ECMAScript Language Specification (2025 edition). §6.1.6.1.3 Number::exponentiate ( base, exponent ). https://tc39.es/ecma262/#sec-numeric-types-number-exponentiate
- IEEE 754-2019. IEEE Standard for Floating-Point Arithmetic. New York: IEEE; 2019. §9.2 Recommended correctly rounded functions — pow. https://doi.org/10.1109/IEEESTD.2019.8766229
- Knuth DE. Two notes on notation. American Mathematical Monthly 1992;99(5):403-422 (the case for 0⁰ = 1). https://doi.org/10.1080/00029890.1992.11995869
- NIST Digital Library of Mathematical Functions, §4.2(iv) Powers — the general power z^a = exp(a ln z) and its principal value, which is why a negative base with a non-integer exponent is complex. https://dlmf.nist.gov/4.2#iv
What can make this go out of date
- None at runtime — the arithmetic and the pow conventions are fixed by the language and floating-point standards; nothing on this page tracks a published series.
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