Hatvány

Hatvány, in mathematics, refers to the exponent in a^b, where a is the base and b the exponent. When b is a positive integer, a^b is the product of b copies of a. For example, 3^4 equals 3 × 3 × 3 × 3. The concept generalizes to other real or complex exponents, but the interpretation and domain depend on the values of a and b.

Notation and basic rules are central to hatvány. For any nonzero a, a^0 = 1. For positive a, a^b is defined for all real b, with a^b interpreted as the b-th power or root depending on b (e.g., a^(1/2) is the square root of a). For negative bases, a^b is defined in the real numbers only for certain rational exponents b = p/q with q odd; otherwise the value is complex. Exponent rules include a^m · a^n = a^(m+n), (a^m)^n = a^(mn), and (ab)^n = a^n b^n for integers n. Negative exponents obey a^(-n) = 1/a^n.

The exponential function and logarithms are closely linked to hatvány. The real exponential function a^x with a > 0 is defined for all real x; its special case e^x (base e) has the distinguished property that d/dx e^x = e^x. The inverse function of a^x for a > 0 is the logarithm log_a(x), with the natural logarithm ln(x) corresponding to the base e. Complex exponentiation extends these ideas beyond the real numbers via complex logarithms and yields multiple values in general.

In Hungarian mathematics, hatvány denotes the exponent, and hatványozás denotes exponentiation. Hatvány concepts underpin growth and decay models, compound interest, physics, and many areas of algebra.