Power Calculator
Raise a base to an exponent: positive, negative, whole or decimal. Whole exponents are calculated exactly, so 2 to the power 64 shows all twenty digits instead of a rounded figure.
Whole numbers, decimals and negatives are accepted.
Result
Enter a base and an exponent to see the power.
The calculation runs in this browser tab. Nothing you enter is uploaded, saved or logged.
What does raising a number to a power mean?
A power is repeated multiplication. In bⁿ, the base b is multiplied by itself n times, so 2⁵ = 2 × 2 × 2 × 2 × 2 = 32. The small raised number is the exponent, and bⁿ is read “b to the power n”.
Exponents extend beyond counting. A zero exponent gives 1 (any non-zero number to the power 0 is 1), a negative exponent means a reciprocal (2⁻³ = 1 ÷ 2³ = 0.125), and a fractional exponent is a root: 9^0.5 is the square root of 9, which is 3.
How to use the Power Calculator
- Enter the base, the number to be multiplied.
- Enter the exponent. It can be negative or have decimals.
- Read the result. Whole exponents give an exact value; other exponents give a value rounded to about 12 significant digits, marked with ≈.
- Very large or very small results also appear in scientific notation. Copy result copies the plain value.
Formula
- Whole exponent
- bⁿ = b × b × … × b (n factors)
- Zero exponent
- b⁰ = 1 (b ≠ 0)
- Negative exponent
- b⁻ⁿ = 1 ÷ bⁿ
- Fractional exponent
- b^(p/q) = the q-th root of b, raised to the power p
Whole exponents are computed on exact fractions, so 1.1² is 1.21 and not 1.2100000000000002 as in many programs.
A fractional exponent such as 0.5 or 0.2 is a root. The calculator checks whether the root is exact (4^0.5 = 2) and says so; otherwise it shows about 12 significant digits.
Worked example
Two to the power 64, the number of values a 64-bit integer can hold. A standard calculator would round this result; here it is exact:
| Question | Result |
|---|---|
| Input: Base | 2 |
| Input: Exponent | 64 |
| 2 to the power 64 | 18,446,744,073,709,551,616 |
| Scientific notation | 1.84467440737 × 10¹⁹ |
| Digits | 20 |
Assumptions and limits
- Whole exponents are limited to ±100,000, and an exact result is limited to 20,000 digits. Beyond that the calculator asks for smaller values instead of freezing your browser.
- Exact results of up to 200 characters are written out in full. Longer ones, and approximate results outside the range 10⁻⁴ to 10¹⁵, are shown in scientific notation with 12 digits, marked ≈ when that is rounded.
- 0⁰ is shown as 1, the usual convention in algebra and combinatorics. It is left undefined in some areas of analysis, and the page says so.
- Zero cannot be raised to a negative power, and a negative base has no real result for exponents with an even root (such as 0.5). The calculator reports these instead of returning a number.
- For a decimal exponent the value comes from floating-point arithmetic rounded to 12 digits; do not rely on the digits beyond that.
Frequently asked questions
What is a negative exponent?
A negative exponent means the reciprocal of the positive power: b⁻ⁿ = 1 ÷ bⁿ. So 10⁻³ = 1 ÷ 1000 = 0.001 and 2⁻² = 0.25.
Why is any number to the power 0 equal to 1?
Because dividing bⁿ by bⁿ gives 1, and by the rules of exponents bⁿ ÷ bⁿ = b⁰. So b⁰ must be 1 for every non-zero b.
What is 0 to the power 0?
By convention 1 in algebra, polynomials and combinatorics, because it keeps formulas such as the binomial theorem working. In analysis it can be left undefined. The calculator shows 1 and notes the convention.
How do I calculate a power with a decimal exponent?
Write the exponent as a fraction: 0.5 = 1/2 is a square root and 0.25 = 1/4 is a fourth root. In general b^(p/q) is the q-th root of b to the power p. For other decimals the value is computed numerically and rounded to about 12 significant digits.
Why can’t I raise a negative number to the power 0.5?
It would be the square root of a negative number, which is not a real number. A negative base works with exponents whose denominator is odd, for example (−8)^(1/3) = −2, and with whole exponents.
Why does the calculator show a long exact number instead of scientific notation?
For whole exponents the result is computed exactly, so it can be shown in full, which is useful for checking digits such as in 2¹⁰⁰. Scientific notation is also given for long results so you can read the magnitude at a glance.