Exponent Calculator
Use this exponent calculator to raise a number to any power, or to work backward. Enter a base and an exponent to get a^b, or give the result and solve for the missing exponent or base. Whole-number powers are computed exactly, digit by digit, so 2^100 shows all 31 digits. Fractions such as 1/3 are read exactly too, which matters for roots of negative numbers.
Updated
Results
- Equation
- Scientific notation
- Exact valueAll digits of a long power, or the fraction for a whole-number base with a negative exponent
- Number of digitsDigits in a whole-number result
- Note
Results are estimates. See the methodology below.
How to use the calculator
- Choose what to solve for. Power finds y = a^b. Exponent finds b when you know a and y. Base finds a when you know b and y.
- Enter the base (a). Decimals and negative numbers are allowed.
- Type the exponent (b). The field accepts whole numbers (10, -3), decimals (2.5) and fractions (1/3 or (2/3)).
- Enter the result (y) when you solve for the exponent or the base.
Scientific notation appears with most power results. The exact value and the digit count appear when they add something: all the digits of a long whole-number power, or the fraction behind a negative exponent.
How the calculator works
The exponent you type is turned into an exact fraction in lowest terms, so 2.5 becomes 5/2 and 0.2 becomes 1/5.
- Whole-number exponent, b ≥ 0: the base is multiplied out exactly as typed, decimals included. That is why 0.1^3 = 0.001, while ordinary floating-point arithmetic gives 0.0010000000000000002. Results up to 30 digits are shown in full; longer ones are shown in scientific form, with every digit (up to 200) in the exact field.
- Negative whole-number exponent: a^(−n) = 1/a^n, shown as a decimal and, for a whole-number base, as an exact fraction.
- Fractional exponent p/q: a^(p/q) is the qth root of a, raised to the power p. For a negative base, an odd q gives a real root. An even q has no real answer, so the calculator shows the principal complex value |a|^b (cos πb + i sin πb).
- Solve for the exponent: x = ln y ÷ ln a, which needs a > 0, a ≠ 1 and y > 0.
- Solve for the base: a = y^(1/b). When b has an even numerator, both a and −a work, so the answer carries a ± sign.
Values that are not exact are shown to 12 significant digits. Results beyond about 10^±300 that can't be multiplied out exactly come from logarithms, to 10 significant digits.
Worked examples
| Input | Result | Note |
|---|---|---|
| 2^10 | 1,024 | 4 digits |
| 1.5^10 | 57.6650390625 | exact decimal |
| 2^-3 | 0.125 | exact: 1/8 |
| (−8)^(1/3) | −2 | real root |
| (−8)^(2/3) | 4 | real root |
| (−4)^0.5 | 2i | complex value |
1. A very large power
2^100 = 1.2676506 × 10³⁰. The exact field shows 1,267,650,600,228,229,401,496,703,205,376, which has 31 digits. For 7^1000 the calculator reports 1.25325664 × 10⁸⁴⁵ and 846 digits.
2. How long does money take to double?
Solve 1.05^x = 2 for the exponent: x = ln 2 ÷ ln 1.05 = 14.2066990829. At 5% compound growth a year, an amount doubles in a little over 14 years. At 7% (1.07^x = 2) it takes 10.2447683511 years. The rule of 72 calculator shows how close the quick mental shortcut gets.
3. Solving for the base
a^2 = 16 gives ±4, because 4² and (−4)² are both 16. a^3 = −27 gives −3. a^(1/2) = 9 gives 81.
Reading the result
A note under the result explains special cases. Real root means a negative base gave a real answer through an odd root. Complex means there is no real answer and you are seeing the principal complex value. No real solution appears when you solve for the base and an even power would have to equal a negative number, as in a^2 = −16.
Frequently asked questions
Why does (−8)^(1/3) give −2 here but an error elsewhere?
The real cube root of −8 is −2, because (−2)³ = −8. Many tools work in floating point, where 1/3 is only an approximation. MDN documents that JavaScript's Math.pow returns NaN for any negative base with a non-integer exponent, even when the real result exists. Other calculators and spreadsheets may return an error or a complex value. This calculator reads 1/3 as an exact fraction, so it can see that the denominator is odd.
Why does typing 0.333 instead of 1/3 change the answer?
0.333 is the fraction 333/1000, and its denominator is even. So (−8)^0.333 has no real value, and the calculator shows the principal complex value, 1.0011190873 + 1.72980323663i. Type 1/3 when you mean a cube root.
What is 0 to the power of 0?
The calculator returns 1 and marks it as a convention. It is a definition that keeps many formulas simple, not a theorem, so a teacher or textbook may treat 0^0 differently. JavaScript's Math.pow(0, 0) also returns 1, according to MDN. Zero to a negative power is undefined, because it would mean dividing by zero, so the calculator shows an error.
What does a negative exponent mean?
It means the reciprocal: a^(−n) = 1/a^n. So 2^-3 = 1/2³ = 1/8 = 0.125. A negative exponent never makes the result negative by itself.
How do I find a square root or cube root with exponents?
Use the exponent 1/2 for a square root and 1/3 for a cube root: 2^(1/2) = 1.41421356237. The square root calculator also shows the simplified radical form, such as √72 = 6√2.
What are the basic exponent rules?
With the same base, multiplying adds exponents (a^m · a^n = a^(m+n)), dividing subtracts them (a^m ÷ a^n = a^(m−n)), and a power of a power multiplies them ((a^m)^n = a^(mn)). Any nonzero number to the power 0 is 1.
Sources
This calculator is a math tool for learning and checking work. Inexact results are rounded to 12 significant digits; check how many digits your class or project requires.