How to use the calculator
Choose what to calculate:
- Power: x^n raises a base x to an exponent n.
- Exponent finds n when you know the base and the result, for example how many times 2 must be multiplied to reach 1,024.
- Base finds x when you know the exponent and the result, which is the same as taking a root.
Enter the numbers and press Calculate. You can type whole numbers, decimals, negative numbers, scientific notation such as 2.5e3, and fractions such as 1/3. The details panel explains each step.
When both the base and the exponent are whole numbers, the result is calculated exactly, digit for digit, as long as it has at most 1,000 digits. Larger or non-whole results are shown with 10 significant digits, in scientific notation when they are very large or very small. For example, 2^10000 is about 1.995063117e+3010, meaning 1.995063117 × 10^3010.
The rules of exponents
- Positive whole exponent: repeated multiplication. 2^10 = 2 × 2 × … × 2 = 1,024.
- Zero exponent: any nonzero number to the power 0 is 1, because x^n ÷ x^n = x^0 = 1.
- Negative exponent: the reciprocal. 2^-3 = 1 / 2^3 = 1/8 = 0.125.
- Fractional exponent: a root. x^(p/q) is the q-th root of x raised to the power p, so 27^(2/3) = 3² = 9.
Solving for the exponent or the base
To find the exponent, take logarithms: if x^n = r, then n = ln r / ln x. For 10^n = 50, n = ln 50 / ln 10 ≈ 1.698970004.
To find the base, take the n-th root: x = r^(1/n). For x² = 49 there are two real answers, 7 and -7, because both squared give 49. The calculator lists both when they exist.
Special cases
- 0^0 is shown as 1. That is the convention used in algebra and combinatorics, where it counts as an empty product. In calculus, limits that look like 0^0 are indeterminate and must be worked out separately.
- Zero to a negative power is undefined, because it would mean dividing by zero.
- Negative base with a fractional exponent: (-8)^(1/3) = -2 is real because the cube root of a negative number exists. When the reduced fraction has an even denominator, as in (-4)^0.5, there is no real result. The calculator says so and shows the principal complex value, here 2i.