Exponent Calculator
Raise any number to any power: negative, fractional like 8^(2/3), decimal or huge exponents, with exact results, simplified radicals and the rule used.
The interactive tool loads here when JavaScript runs.
About this toolWhat is the Exponent Calculator?
Type a base and an exponent and the calculator works out bⁿ with the rule written out: a negative exponent becomes a reciprocal, a fractional exponent becomes a root, and a whole-number exponent is multiplied out exactly. 2^−3 is 1/8, 8^(2/3) is 4, 2^(3/2) is 2√2 and (2/3)^5 is 32/243.
Whole-number powers are computed with big integers, so 2^1000 appears with all 302 digits and 1.05^30 appears as its exact 60-digit decimal, not a rounded approximation. When the answer is too large to print, such as 2^136279841, the page still gives the number of digits, the leading digits from an 80-digit logarithm and the last 20 digits from modular arithmetic.
Irrational answers come out as a simplified radical where one exists and as a 30-digit decimal, and e and pi are accepted as bases for exponential growth problems.
- Negative, fractional, decimal and very large exponents
- Exact integers and fractions using big-integer arithmetic
- Simplified radical form for irrational results
- Real roots of negative bases with odd denominators
- Digit count and exact last digits for enormous powers
- Bases e and pi
How to use the Exponent Calculator
- Type the base. It can be a whole number, decimal, fraction like 2/3, a negative number, e or pi.
- Type the exponent. Use a fraction such as 2/3 for roots; 0.667 is treated as 667/1000 and gives a different answer.
- Read the result. Exact answers are labelled exact; irrational ones show a radical form and 30 significant digits.
- For long answers, use Copy all to copy every digit, or read the digit count and last digits for enormous powers.
- Open the steps to see which exponent rules were applied.
Worked examples
Fractional exponent
Input
8^(2/3)
Output
4
The cube root of 8 is 2, and 2² = 4. Typing 2/3 keeps it exact.
Negative exponent
Input
2^−3
Output
1/8 = 0.125
A negative exponent means the reciprocal of the positive power: 1 / 2³.
Compound growth
Input
1.05^30
Output
4.321942375150662009157288198886473341473378241062164306640625
Thirty years at 5% multiplies money by about 4.32. Because 1.05 is a terminating decimal, the exact answer terminates too.
A record prime
Input
2^136279841
Output
41,024,320 digits, starting 8.8169…
One less than this power is the largest known prime as of October 2024. Its digit count is floor(136279841 × log₁₀2) + 1.
Background and insight
An exponent counts repeated multiplication: bⁿ = b × b × … × b with n factors. The other rules follow from keeping the pattern bᵐ × bⁿ = bᵐ⁺ⁿ consistent. Setting n = 0 forces b⁰ = 1 for any non-zero b. Setting m = −n forces b⁻ⁿ = 1/bⁿ. Asking what squared gives b forces b^(1/2) = √b, and in general b^(p/q) = (q-th root of b)^p.
Negative bases are where calculators disagree. (−8)^(1/3) has the real answer −2 because cube roots of negatives are real, yet many programming languages return NaN because they compute powers as exp(n × ln b), and the log of a negative number is not real. This calculator reduces the exponent to lowest terms first and returns the real root whenever the denominator is odd; with an even denominator, as in (−8)^(1/2), no real answer exists and the principal complex value is shown.
0⁰ is set to 1 here, matching the convention in algebra, combinatorics and the IEEE 754 pow function, with a note that it is an indeterminate form when it arises as a limit. 0 raised to a negative power is a division by zero and is reported as undefined.
Common use cases
Exponent rules homework
Check negative and fractional exponents and see the rule that turns each into a reciprocal or a root.
Interest and growth
Raise 1 plus a rate to a number of periods to find a growth factor, exactly rather than to eight decimals.
Computing and number theory
See the digit count and last digits of huge powers of 2, 3 or 10 used in hashing, key sizes and puzzles.
Science and engineering
Evaluate power laws and e^x terms with enough digits that rounding never hides a difference.
Tips and limitations
- Enter roots as fractions (1/3), not decimals (0.333). The decimal is a slightly different exponent and loses the exact answer.
- Brackets matter on paper: −2² is −4 while (−2)² is 4. Here the base field is always the whole base, so −2 means (−2).
- A number with more than 100,000 digits is summarised by its digit count, leading digits and last digits instead of being printed.
- For logarithms, the inverse of exponents, use the Logarithm Calculator.
Frequently asked questions
What does a negative exponent mean?
It means take the reciprocal. b⁻ⁿ = 1 / bⁿ, so 2⁻³ = 1/8 and 10⁻² = 0.01. The exponent rules force this: 2³ × 2⁻³ must equal 2⁰ = 1.
How do you work out a fractional exponent?
The denominator is a root and the numerator is a power: b^(p/q) = (q-th root of b)^p. 8^(2/3) = (∛8)² = 2² = 4, and 16^(3/4) = (⁴√16)³ = 2³ = 8.
What is 0 to the power 0?
In algebra, combinatorics and most programming languages it is defined as 1, and the calculator uses that. In calculus, an expression that approaches 0⁰ is an indeterminate form whose limit depends on the functions involved.
Why does my calculator say (−8)^(1/3) is an error?
Many calculators and languages compute powers through logarithms, which fail for negative bases. The real cube root of −8 is −2, which this calculator returns because the exponent's denominator, 3, is odd.
How do I find how many digits a big power has?
The digit count of bⁿ is floor(n × log₁₀ b) + 1. For 2^1000 that is floor(301.03) + 1 = 302 digits. The calculator does this with an 80-digit logarithm so the count is exact even for exponents in the billions.
Can the result be exact for decimals?
Yes, for whole-number exponents. A decimal like 1.05 is the fraction 21/20, so any whole power of it is a fraction with a power of 10 at the bottom, which always ends. 1.05^30 has exactly 60 decimal places.
What is the difference between −2² and (−2)²?
Order of operations applies the exponent before the minus sign, so −2² = −(2²) = −4, while (−2)² = 4. In this calculator the base box holds the whole base, so typing −2 means (−2).
Is there an in-depth guide to the Exponent Calculator?
Yes. “Exponent Rules: Negative, Zero and Fractional Powers Explained” covers the background, how it works, common pitfalls and worked examples. Read the full guide.
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