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Exponent Calculator
Enter a base and an exponent to compute the power. Whole, negative and fractional exponents are supported, with exact integer results where possible and clear messages when no real answer exists.
What an exponent means
An exponent says how many times to use the base as a factor. In bn with n a positive whole number, you multiply b by itself n times, so 210 = 1,024. The rest of the definitions extend this idea in a way that keeps the rules consistent.
- Zero exponent: any nonzero base to the power 0 is 1. This follows from bn ÷ bn = 1.
- Negative exponent:
b−n = 1 ÷ bn, so 2−3 = 1/8 = 0.125. - Fractional exponent:
b1/nis the n-th root, andbm/nis the n-th root of b raised to m. So 81/3 = 2 and 82/3 = 4.
The laws of exponents
These identities hold whenever both sides are defined for real numbers:
| Law | Example |
|---|---|
| bm × bn = bm+n | 23 × 24 = 27 |
| bm ÷ bn = bm−n | 56 ÷ 52 = 54 |
| (bm)n = bmn | (32)3 = 36 |
| (ab)n = anbn | (2 × 5)2 = 4 × 25 |
Note that bm + bn has no simple rule. Adding powers does not combine the exponents.
Negative bases and when there is no real answer
A negative base works with whole exponents: (−2)3 = −8 and (−2)4 = 16, so an odd exponent keeps the sign and an even one removes it. With a fractional exponent the answer depends on the denominator of the fraction. An odd denominator is fine, so (−8)1/3 = −2. An even denominator, such as (−4)1/2, would need the square root of a negative number, which is not real. This page reports that case instead of guessing, and the square root calculator shows the imaginary form.
Zero to a negative power is undefined because it requires dividing by zero. Zero to the power zero is sometimes defined as 1 for convenience, and this page returns 1 for it.
Precision and large results
Results from the browser use double-precision floating point, which holds about 15 to 16 significant digits. When both base and exponent are whole numbers and the exponent is not negative, the page also computes the exact integer with big-integer arithmetic, so 2100 is shown as 1267650600228229401496703205376 rather than a rounded value. Beyond 40 digits the display shows the leading and trailing digits and the digit count. Results above about 10308 overflow and are reported as too large. Scientific notation is shown for very large or very small values; see the scientific notation calculator to convert numbers in either direction.
Worked example
Compute 2 to the power 100, a number so large that the exact digits matter more than a rounded decimal.
Inputs
| Base | 2 |
|---|---|
| Exponent | 100 |
Result
Base 2 raised to the power 100.
| Result | 1.2676506e30 |
|---|---|
| Scientific notation | 1.267651 × 10^30 |
| Exact integer | 1267650600228229401496703205376 |
| Number of digits | 31 |
Powers of 2
| Exponent n | 2^n |
|---|---|
| 0 | 1 |
| 1 | 2 |
| 2 | 4 |
| 3 | 8 |
| 4 | 16 |
| 5 | 32 |
| 6 | 64 |
| 7 | 128 |
| 8 | 256 |
| 9 | 512 |
| 10 | 1,024 |
Accuracy note: Results use standard double-precision arithmetic, so very large or very small values may be rounded. Check important results by another method.
Frequently asked questions
What does a negative exponent mean?
It means take the reciprocal of the positive power. For example, 2 to the power −3 equals 1 divided by 2 cubed, which is 1/8 or 0.125.
Why is any number to the power zero equal to 1?
Dividing a power by itself gives b^n ÷ b^n = 1, and the exponent law for division subtracts the exponents to give b^0. So b^0 must equal 1 for any nonzero base.
What is a fractional exponent?
The denominator of the fraction is a root and the numerator is a power. So 27^(2/3) means the cube root of 27, which is 3, squared, giving 9.
Why does a negative base with a decimal exponent give an error?
Roots with an even index of negative numbers are not real. The result would be complex, and this calculator only returns real numbers.
How large an exponent can I use?
Results up to about 1.8 × 10^308 can be shown. Exact integers are computed for whole-number bases up to a million and exponents up to 2,000.
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References and sources
- NIST Digital Library of Mathematical Functions, section 4.2: definitions of powers and exponentials. Link
- Stewart, J., Precalculus: Mathematics for Calculus, chapter on exponents and radicals.