exponentiates

Exponentiate is a mathematical operation that raises a quantity to a power. In the common notation a^b, the base a is conceptually multiplied by itself b times when b is a positive integer; for real exponents the interpretation extends via the exponential and logarithmic functions, and when the base is the number e the expression e^b denotes the natural exponential function. The verb exponentiate is used to describe this operation, and the third-person singular present form is exponentiates.

Key rules and conventions include: a^m a^n = a^{m+n} (for a > 0, or when m and n are integers for any nonzero a); (a^m)^n = a^{mn}; (ab)^m = a^m b^m under appropriate conditions (typically for nonnegative bases or integer exponents). Negative exponents satisfy a^{-n} = 1/a^n (a ≠ 0), and a^0 = 1 (a ≠ 0). Fractional exponents correspond to roots: a^{p/q} = q-th root of a^p, defined for a ≥ 0 when q is even; with a > 0 these laws extend to real exponents.

In the complex setting, a^z is defined as exp(z Log a), where Log is the complex logarithm; this expression is multivalued due to the logarithm’s multiple values. The exponential function e^x (the base e) is smooth and has derivative equal to itself, making it fundamental in calculus and analysis. Exponentiation models exponential growth and decay in natural phenomena and finance, and serves as the inverse operation to the logarithm.