Solve for base (a), exponent (n), or power result (y) in the exponential equation aⁿ = y, with support for negative powers, decimals, and arbitrary-precision integer calculations.
Exponentiation is a mathematical operation written as aⁿ, involving two numbers: the base a and the exponent or power n. When n is a positive integer, exponentiation corresponds to repeated multiplication of the base:
When multiplying powers with the same base, add exponents:
aᵐ · aⁿ = aᵐ⁺ⁿWhen dividing powers with the same base, subtract exponents:
aᵐ / aⁿ = aᵐ⁻ⁿMultiply the inner and outer exponents:
(aᵐ)ⁿ = aᵐ·ⁿDistribute exponent to every factor in the product:
(ab)ⁿ = aⁿ · bⁿAny non-zero base raised to the power of zero equals 1:
a⁰ = 1 (for a ≠ 0)A negative exponent is the reciprocal of the positive power:
a⁻ⁿ = 1 / aⁿFractional powers correspond directly to radical roots:
aᵐ/ⁿ = ⁿ√(aᵐ) = (ⁿ√a)ᵐIn calculus and analysis, 0⁰ is classified as an indeterminate form because the limit depends on the path taken. However, in discrete mathematics, set theory, combinatorics, and computer programming, 0⁰ is universally defined as 1 (for example, to ensure polynomials and binomial series work without exception).
The solver uses logarithms. By taking the natural logarithm of both sides of aⁿ = y, we get n · ln(a) = ln(y), which rearranges to n = ln(y) / ln(a) (the change of base formula).
Raising a negative number to a fractional exponent (such as (-4)^0.5) requires taking an even root of a negative number (√(-4)), which produces an imaginary/complex number (2i). This calculator evaluates operations strictly within the real number system.
A negative exponent indicates division rather than multiplication. For example, 2⁻³ = 1 / 2³ = 1 / 8 = 0.125.