How to use
- Enter the base. Typing −2 means the base is −2, as in (−2)ⁿ.
- Enter the exponent: a whole number (3), a negative number (−2), a decimal (0.5) or a fraction (2/3).
- Read the result, its scientific notation, and the steps. Big whole-number powers show every digit.
Exponent rules and formulas
An exponent says how many times to use the base as a factor: 2⁵ = 2 × 2 × 2 × 2 × 2 = 32.
| Rule | Example | Result |
|---|---|---|
| Zero exponent: a⁰ = 1 | 9⁰ | 1 |
| Negative exponent: a⁻ⁿ = 1/aⁿ | 4⁻² | 1/16 = 0.0625 |
| Fraction: a^(m/n) = (ⁿ√a)^m | 125^(2/3) | 25 |
| Product: aᵐ × aⁿ = aᵐ⁺ⁿ | 2³ × 2⁵ = 2⁸ | 256 |
| Power of a power: (aᵐ)ⁿ = aᵐⁿ | (2³)² = 2⁶ | 64 |
A negative base with a fractional exponent has a real answer only when the root is odd: (−8)^(1/3) = −2, but (−16)^(1/4) is not a real number.
Examples
- 2¹⁰: ten 2s multiplied together give 1,024.
- 4⁻²: 1 ÷ 4² = 1 ÷ 16 = 0.0625.
- 16^(−3/2): √16 = 4, then 4³ = 64, then 1 ÷ 64 = 0.015625.
- 1.05¹⁰: about 1.628894627, the growth factor of 5% a year for 10 years.
- 2⁶⁴: exactly 18,446,744,073,709,551,616, about 1.844674407 × 10¹⁹.
0⁰, −2² and very large powers
0⁰ is shown as 1, the usual convention in software. Algebra courses define a⁰ = 1 only for non-zero a, and in calculus 0⁰ is an indeterminate form.
The power applies before a minus sign: −2² = −(2²) = −4, while (−2)² = 4.
Whole-number powers are calculated exactly. Results too large for ordinary numbers, such as 2¹⁰⁰⁰⁰ ≈ 1.995063117 × 10³⁰¹⁰, are still shown in scientific notation.
Square and cube roots work the other way round; see the square root calculator. For growth over time, try the compound interest calculator.