Exponent Calculator

Background.

Powers turn up whenever something multiplies by the same factor again and again. Money earning 7% a year is multiplied by 1.07 each year, so after a decade it has grown by 1.07¹⁰ = 1.9672 — almost exactly double. A radioactive sample keeps half of what it had every half-life, so after three half-lives 0.5³ = 0.125 of it remains. Ten binary digits can count 2¹⁰ = 1,024 different values, which is where the computer kilobyte of 1,024 bytes came from.

Each of those is one equation, aⁿ = x, and in practice you can be missing any of its three numbers. Usually it is the result: you know the base a and the exponent n and want x. Sometimes it is the base: what yearly growth factor doubles a sum in ten years? That is the tenth root of 2, about 1.0718, or 7.18% a year. And often it is the exponent: how many years at 5% does doubling take? That is a logarithm, log base 1.05 of 2, which comes to 14.2067 years. The solve-for menu chooses which of the three you want, and the remaining two fields hold what you already know.

The exponent does not have to be a positive whole number. A negative exponent means a reciprocal, so 2⁻³ = 1/8 = 0.125. A fraction means a root, so 16^0.25 is the fourth root of 16, which is 2. Any other decimal, such as 4^1.5 = 8, is handled the same way, through the logarithm definition of a power.

A few combinations have no real-number answer, and you get a message instead of a number. A negative base with a fractional exponent is the usual one: (−8)^0.5 would be a square root of a negative number. Zero raised to a negative power would mean dividing by zero. And results beyond about 1.8 × 10³⁰⁸ are larger than standard computer arithmetic can hold, so 10³⁰⁹ is declined while 10³⁰⁸ is shown. Alongside every answer you also get its scientific-notation form, a mantissa between 1 and 10 and a power of ten, which is the only comfortable way to read a figure such as 2¹⁰⁰ ≈ 1.268 × 10³⁰.

What is exponent calculator?

Exponentiation is repeated multiplication extended to every kind of exponent. For a positive whole number n, aⁿ means n copies of a multiplied together, so 5³ = 5 × 5 × 5 = 125. The definition is then stretched so that the index laws keep working: a⁰ = 1 for any non-zero a, a⁻ⁿ = 1/aⁿ, and a^(1/n) is the nth root of a, the number whose nth power gives back a. For a positive base and any real exponent, aⁿ is defined as e^(n·ln a), which agrees with all the cases above and fills in the gaps between them. The operation has two inverses, because the base and the exponent play different roles: the nth root recovers the base from the result, and the logarithm recovers the exponent. Those are the three directions the solve-for menu offers.

How to use this calculator.

1. 01Choose what to solve for. Result computes aⁿ from a base and an exponent; Base finds the number that, raised to your exponent, gives your result; Exponent finds the power that turns your base into your result. Whichever field you are solving for is ignored, so you can leave its old value in place.
2. 02For Result, enter the base and the exponent. Type fractional exponents as decimals: 0.5 for a square root, 0.25 for a fourth root, 1.5 for 'square root, then cubed'. Put a negative base in the Base field as a negative number; (−2) squared is entered as base −2, exponent 2, and gives 4.
3. 03For Base, enter the known result and the exponent that produced it. A square root uses exponent 2, not 0.5: result 9 with exponent 2 returns 3, while exponent 0.5 would return 81, the number whose square root is 9. Negative results work with odd exponents (result −8, exponent 3 gives −2) and are declined with even ones.
4. 04For Exponent, enter a base and a result, both above zero, with the base not equal to 1. Base 10 gives the common logarithm, base 2 the number of doublings, and base 2.718281828459045 (e) the natural logarithm. A base between 0 and 1 is allowed and counts halvings or other shrinking steps.
5. 05Read the scientific-notation pair under the main answer. It always describes the Result value, so in Base and Exponent modes it restates the result you typed rather than the number you solved for.

The formula.

Each mode rearranges x = aⁿ, where a is the base, n the exponent and x the result. In Result mode the calculator evaluates aⁿ directly. In Base mode it computes a = x^(1/n), the nth root of x; when x is negative and n is an odd whole number it takes the root of |x| and makes the answer negative, because (−2)³ = −8 means the cube root of −8 really is −2. With n even, or not a whole number, a negative x has no real root and is refused, and n = 0 is refused because every non-zero base gives a⁰ = 1, so the base cannot be recovered. In Exponent mode it uses the change-of-base rule n = ln x ÷ ln a, which needs a > 0, a ≠ 1 and x > 0; base 1 would put ln 1 = 0 in the denominator. The scientific-notation outputs come from the result: the power of ten is the whole-number part of log₁₀|x| rounded down, and the mantissa is x divided by that power of ten, so 1,024 becomes 1.024 × 10³. The index laws all follow from the repeated-multiplication picture: aᵐ·aⁿ = aᵐ⁺ⁿ, aᵐ/aⁿ = aᵐ⁻ⁿ, (aᵐ)ⁿ = aᵐⁿ, (ab)ⁿ = aⁿbⁿ and (a/b)ⁿ = aⁿ/bⁿ. For example 2¹⁰ × 2³ = 2¹³ = 8,192, while (2³)² = 2⁶ = 64. A fractional exponent m/n combines a root and a power in either order, so 27^(2/3) is the cube root of 27, which is 3, squared to give 9. Because the exponent multiplies the logarithm of the base, a long run of periods magnifies small differences in the base: 1.07³⁰ = 7.6123 but 1.08³⁰ = 10.0627, about a third more from a single extra percentage point of growth. Numbers are carried in standard double-precision arithmetic and shown to 15 significant figures, which is far more than everyday work needs but not enough to spell out every digit of very large integers.

A worked example.

Frequently asked questions.

References& sources.

1. [1]Knuth, Donald E. The Art of Computer Programming, Volume 1: Fundamental Algorithms, 3rd edition. Addison-Wesley, 1997 — §1.2.4 (Integer Functions and Elementary Number Theory) and §1.2.5 (Permutations and Factorials) cover the formal definition of integer exponentiation, the index laws, and the algorithmic complexity of fast exponentiation (binary method, O(log n)).
2. [2]Hardy, G. H. and Wright, E. M. An Introduction to the Theory of Numbers, 6th edition. Oxford University Press, 2008 — §1.4 develops integer powers and the index laws from first principles, and §4.1–4.3 extend to rational and real exponents via the limit definition.
3. [3]NIST Digital Library of Mathematical Functions, §4 — Elementary Functions, particularly §4.2 (Definitions) and §4.4 (Logarithms and Powers). The definitive modern reference for the analytic definition of the power function a^z for real and complex arguments, the index laws, and numerical evaluation strategies.
4. [4]Abramowitz, M. and Stegun, I. A. (eds.) Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. National Bureau of Standards Applied Mathematics Series 55, 1964 — §4 (Elementary Transcendental Functions), particularly §4.2 on power functions and §4.1 on the exponential and logarithm. Still cited by NIST DLMF as a foundational reference.
5. [5]Lang, Serge. Algebra, 3rd edition. Graduate Texts in Mathematics 211, Springer, 2002 — Chapter 1 develops exponentiation in arbitrary monoids and groups, the index laws as group-theoretic identities, and the extension from integer to rational exponents via the structure of ordered fields.
6. [6]ISO 80000-2:2019 — Quantities and units, Part 2: Mathematics. International standard fixing the notation a^n for exponentiation, n√a for the nth root, and the constraints on real-valued evaluation (positive base for non-integer exponents).