Exponentiation

Exponentiation is a mathematical operation that raises a base to a given exponent. It is written as a^b, where a is the base and b is the exponent. For positive integer exponents n, a^n equals the product of n copies of a: a^n = a · a · ... · a (n factors).

Key special cases include a^0 = 1 for a ≠ 0 and a^{-n} = 1/a^n for n > 0. If a is zero, 0^0 is indeterminate in many contexts, and 0^(-n) is undefined. Exponent laws hold: a^m a^n = a^{m+n}, (a^m)^n = a^{mn}, and a^m / a^n = a^{m-n} for a ≠ 0. Fractional and real exponents extend the notion: a^{p/q} is the qth root of a^p, defined when appropriate (for real values often a ≥ 0 and q is odd or a real qth root exists). In general, real-valued exponentiation is defined for a > 0 by a^b = e^{b ln a}, which yields real results for any real b; negative bases require complex values for non-integer exponents.

Special case base e: e^x is the natural exponential function, fundamental in mathematics. It satisfies d/dx e^x = e^x, and in general d/dx a^x = a^x ln a for a > 0.

Applications include growth and decay models, compound interest, and analysis. Exponentiation is inverse to logarithms: log_b(a) is the exponent to which b must be raised to obtain a. The exponential function underpins many areas of science and engineering.