Exponent Calculator

Exponentiation — raising a base to a power — is shorthand for repeated multiplication. The expression 2⁵ means 2 × 2 × 2 × 2 × 2 = 32. Here, 2 is the base and 5 is the exponent (or index). Exponents appear throughout mathematics, science, and computing: compound interest, population growth, computer memory sizes (2¹⁰ = 1,024 bytes in a kilobyte), and scientific notation all rely on powers.

Key exponent rules simplify complex expressions. When multiplying powers with the same base, add the exponents: xᵃ × xᵇ = x^(a+b). When dividing, subtract: xᵃ ÷ xᵇ = x^(a−b). When raising a power to another power, multiply: (xᵃ)ᵇ = x^(a×b). Any nonzero number raised to the zero power equals 1: 5⁰ = 1, (−3)⁰ = 1. These rules follow from the definition of exponents and are essential for algebra and calculus.

Negative exponents represent reciprocals: x^(−n) = 1/xⁿ. So 2^(−3) = 1/2³ = 1/8 = 0.125. This definition ensures the exponent rules work consistently for all integers. Fractional exponents connect to roots: x^(1/n) = ⁿ√x, so 8^(1/3) = ∛8 = 2. The expression x^(m/n) means the nth root of x, raised to the mth power.

When the base is negative, the result depends on whether the exponent is even or odd. (−2)⁴ = 16 (even exponent, positive result) but (−2)³ = −8 (odd exponent, negative result). Non-integer exponents of negative bases produce complex numbers, which this calculator does not display. Very large exponents can exceed the range of standard floating-point numbers, producing overflow.

Practical applications include calculating compound interest (A = P(1 + r)ᵗ), estimating computational complexity (O(2ⁿ) algorithms), converting between units in scientific notation, and modeling exponential growth and decay in biology and physics. Enter any base and exponent to compute the power instantly — the calculator handles positive, negative, and zero exponents for valid real-number results.