How the Exponent Calculator Works
An exponent says how many times to multiply a number by itself. In 28, the number 2 is the base and 8 is the exponent. It means 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 = 256. The Exponent Calculator raises any number to any power, including negative and fractional powers.
What you enter
- Base (b): the number to multiply (the default is 2).
- Exponent (x): the power (the default is 8). It can be negative, a decimal or zero.
The result
The calculator shows b^x, and also the same value in scientific notation. Whole numbers are shown with every digit, even when they are very big. For example, 2^100 is shown as 1,267,650,600,228,229,401,496,703,205,376. Extremely large or small answers are shown in scientific notation, such as 1.5e+20, instead of a long string of digits.
The rules of exponents
| Rule | Example |
|---|---|
| Exponent 0 gives 1 | 7^0 = 1 |
| Exponent 1 gives the number | 7^1 = 7 |
| Negative exponent means 1 / the power | 2^-3 = 1/8 = 0.125 |
| Exponent 0.5 is the square root | 9^0.5 = 3 |
| Same base, add the exponents | 2^3 x 2^4 = 2^7 = 128 |
| Power of a power, multiply | (2^3)^2 = 2^6 = 64 |
Worked examples
| Expression | Result |
|---|---|
| 2 ^ 8 | 256 |
| 2 ^ 10 | 1,024 |
| 5 ^ 3 | 125 |
| 10 ^ -2 | 0.01 |
| (-3) ^ 3 | -27 |
| 2 ^ 0.5 | 1.4142135624 |
| 2 ^ -10 | 0.0009765625 |
Chart: powers of 2
Each step doubles the number, so the bars grow quickly:
2^1 █ 2 2^2 ███ 4 2^3 █████ 8 2^4 ██████████ 16 2^5 ████████████████████ 32 2^6 ████████████████████████████████████████ 64
Chart: powers of 2 and powers of 3
The blue bars are powers of 2 and the green bars are powers of 3. A bigger base grows faster:
n=1 █ 2 █ 3 n=2 ██ 4 ████ 9 n=3 ████ 8 █████████████ 27 n=4 ████████ 16 ████████████████████████████████████████ 81
When there is no answer
- A negative base with a fractional exponent, such as (-4)^0.5, has no real-number answer.
- 0 to a negative power means dividing by zero, so it is undefined.
- Too large: a result beyond what a computer can hold is rejected.
Where exponents are used
- Growth: compound interest and population growth use powers.
- Computers: sizes double each time, so they are powers of 2 (1 KB = 2^10 bytes).
- Science: very large and very small numbers are written with powers of 10, like 6 x 10^23.
- Geometry: area uses a square (^2) and volume uses a cube (^3).
Tips
- A negative base needs brackets to read correctly: (-2)^2 is 4, but -2^2 means the negative of 2^2, which is -4.
- Even powers of a negative number are positive, and odd powers are negative.
This tool gives an estimate for general information only. Decimal answers are shown with up to 10 decimal places.