Potensreglen

Potensreglen refers to a set of algebraic rules governing exponentiation used to simplify and manipulate expressions that involve powers. These rules are foundational in algebra and are applied across mathematics to handle products, quotients, and powers of variables and constants. They are valid under certain conditions on the base, typically requiring the base to be nonzero for negative exponents and zero exponents to be treated carefully.

Key rules include the product rule: a^m · a^n = a^{m+n} for a ≠ 0, which allows combining powers with the same base. The quotient rule states that a^m / a^n = a^{m−n} for a ≠ 0. The power rule says (a^m)^n = a^{mn}, with the usual caveats for real exponents: for real m and n, a > 0 ensures a^m is defined; for integer exponents, the base may be nonzero. Negative exponents follow a^{-n} = 1 / a^n, provided a ≠ 0, and any nonzero base satisfies a^0 = 1.

Other rules cover powers of products and quotients: (ab)^n = a^n b^n and (a/b)^n = a^n / b^n, with b ≠ 0. Fractional exponents are interpreted as a^{p/q} = (q-th root of a)^p, defined for a ≥ 0 in the real numbers; extensions to negative bases require additional considerations, such as complex numbers.

In practice, potensreglen facilitates simplification, expansion, and solving of equations, and it underpins many methods in calculus and analytic proofs. See also exponentiation, logarithms, and related algebraic rules.