exponierte

Exponierte is a term used in mathematics to denote the result of exponentiation, the operation of raising a quantity to a power. In several languages, including German and Dutch, the verb exponieren or exponeren yields participles such as exponierte to describe a value that has been raised to a power.

Notation and definitions: Exponentiation is written as a^b, where a is the base and b is the exponent. If b is a real number, a must be positive for a^b to be real; otherwise the expression can be extended to complex values. For a>0, a^b can be defined as e^{b ln a}. When b is an integer, exponentiation is the product of a with itself b times; negative exponents yield a^−n = 1/(a^n); fractional exponents represent roots, as in a^{m/n} = the n-th root of a^m, when defined in the real numbers.

Relation to the exponential function: The exponential function e^x is a related but distinct concept, defined as the power function with base e. It is the inverse of the natural logarithm and plays a central role in growth and decay models as well as in calculus.

Properties and limitations: For a>0, (a^b)^c = a^{bc}, and a^b · a^c = a^{b+c}. Exponentiation is not generally commutative, since a^b ≠ b^a except in special cases. Non-integer exponents introduce subtleties about branches when extending to complex numbers, and the zero base with non-positive exponents is undefined in the real numbers.

Applications and relevance: Exponentiation underpins compound interest, population growth, exponential models, and many algorithms in computer science. The concept of exponierte values remains central in fields ranging from finance to physics and engineering.