Transparent method
The formula
Worked example
Example inputs
How to interpret the result
A negative exponent and a fractional exponent get treated as two unrelated edge cases, when they're really the same rule applied twice. A negative exponent means take the reciprocal, and a fractional exponent means take a root. Combine the two and an expression like a base raised to negative one-half means the reciprocal of that base's square root, not a small negative number, which is the answer most people guess first.
Before you rely on it
What to check
Split a combined exponent, negative and fractional together, into its two separate operations, reciprocal and root, before trusting an intuition about the sign or size of the result.
The common error
Where people go wrong with exponent calculator
Assuming a negative exponent always produces a negative result. It produces a reciprocal, which stays positive for a positive base: the sign of the exponent controls magnitude, not the sign of the answer.
Sensitivity evidence
How base changes the result
| Base | Result | Digits before the decimal point |
|---|---|---|
| 1.6 | 110 | 3 |
| 1.8 | 357 | 3 |
| 2worked example | 1,024 | 4 |
| 2.2 | 2,656 | 4 |
| 2.4 | 6,340.3 | 4 |
Every input, tested
Which input moves the result most
| Input | Tested from | To | Result at each end | Swing |
|---|---|---|---|---|
| Base | 1.8 | 2.2 | 357 to 2,656 | 2,298.9 (225%) |
| Exponent | 9 | 11 | 512 to 2,048 | 1,536 (150%) |
Two variables at once
Result by base and exponent
| Base \ Exponent | 8 | 10 | 12 |
|---|---|---|---|
| 1.6 | 42.9 | 110 | 281.5 |
| 1.8 | 110.2 | 357 | 1,156.8 |
| 2 | 256 | 1,024 | 4,096 |
| 2.2 | 548.8 | 2,656 | 12,855 |
| 2.4 | 1,100.8 | 6,340.3 | 36,520.3 |
Step by step
The worked example, input by input
| Input | Value used |
|---|---|
| Base | 2 |
| Exponent | 10 |
| Result | 1,024 |
| Digits before the decimal point | 4 |
Where the inputs come from
Enter the base, the number being multiplied, and the exponent, how many times. The calculator handles positive, negative and fractional exponents. A negative exponent means the reciprocal, so 2 to the power of minus 3 is 1 over 8. A fractional exponent means a root, so 9 to the power of one half is 3. Zero raised to a negative power is undefined, and a negative base with a fractional exponent may have no real answer.
When the result stops applying
Results can overflow the display for large exponents, and floating-point rounding affects long decimals. Any nonzero number to the power of zero is 1, and zero to the power of zero is treated differently by different conventions. Order of operations matters: negative signs and exponents combine so that minus 2 squared is different from the square of minus 2.
What to compare it with
See roots on the root calculator, logarithms on the logarithm page, and scientific notation for very large numbers. The exponential growth calculator applies exponents to change over time, and the compound interest calculator applies them to money. The scientific calculator handles more complex expressions.
People also ask
Frequently asked questions
What does a negative exponent mean?
It means the reciprocal of the positive power, so 2 to the minus 3 equals 1 divided by 2 cubed, or one eighth.
What is any number to the power of zero?
One, for any nonzero base. This follows from the rule that dividing equal powers gives one.
What does a fractional exponent mean?
A root. An exponent of one half is a square root, and one third is a cube root.