exponentiatie

Exponentiatie, in English exponentiation, is a mathematical operation that raises a base to a given exponent. If the base is a and the exponent is b, the expression a^b represents the base multiplied by itself b times when b is a positive integer. For b = 0, a^0 = 1 (a ≠ 0). If b is negative, a^b = 1/a^{|b|}. For real exponents to yield a real number, the base is typically required to be positive; with a ≤ 0, exponentiation is defined only for certain rational exponents and is not real in general.

Notation and basics extend beyond positive integers. When a > 0 and b is any real number, a^b is well defined via the exponential function and logarithms. If the exponent is a rational number p/q in lowest terms, a^{p/q} can be interpreted as the q-th root of a^p, provided a ≥ 0. The expression (-8)^{1/3} = -2 is an example of a real cube root.

Key rules govern exponentiation. For bases a (a ≠ 0) and exponents m and n, a^m a^n = a^{m+n}, and (a^m)^n = a^{mn}. If c is an integer, (ab)^c = a^c b^c; for positive bases with real exponents, (ab)^c = a^c b^c also holds when a > 0 and b > 0.

The exponential function with base e, written e^x, is central in many contexts; its inverse function is the natural logarithm, ln x. Exponentiation is used to model growth and decay, compound interest, and many equations involving exponential change.

Examples: 2^3 = 8; 4^{1/2} = 2; e^2 ≈ 7.389; (-8)^{1/3} = -2.