PotenzWurzelgesetze

PotenzWurzelgesetze are fundamental algebraic rules that govern the manipulation of exponentiation and radical expressions. They provide a systematic way to simplify, combine, and transform powers and roots, which is essential for solving equations, proving identities, and working in many areas of mathematics, physics, and engineering.

The most common laws involve the following operations: multiplication of powers with the same base increases the exponent; division of powers with the same base decreases the exponent; raising a power to another power multiplies the exponents; and raising a power to the power of zero yields one, provided the base is non‑zero. For radicals, the product of roots with the same radicand can be combined by adding the indices, and a root of a power can be simplified by dividing the exponent by the root index. The inverse relationships between exponents and radicals are also captured: a root can be expressed as a fractional exponent, and a fractional exponent can be rearranged into a radical.

These laws are often written in symbolic form: a^m × a^n = a^{m+n}; a^m ÷ a^n = a^{m−n}; (a^m)^n = a^{mn}; √[n]{a^m} = a^{m/n}. They hold for real numbers a ≥ 0 when n is a positive integer, and are extended to complex numbers with appropriate branch cuts. In addition, when working with negative bases for even-indexed roots, restrictions apply to maintain real results.

PotenzWurzelgesetze provide a compact language for manipulating algebraic expressions involving exponents and radicals. Mastery of these rules allows students and professionals to rewrite expressions in simplified forms, verify identities, and solve polynomial equations efficiently.