potensform

Potensform is a mathematical representation that expresses numbers or expressions as a base raised to a power. In this form, a quantity is written as a^b, where the base a is the number being multiplied by itself, and the exponent b indicates how many times. In school contexts, bases are typically positive real numbers and exponents are nonnegative integers, though more general definitions include negative and fractional exponents.

Examples include 16 = 2^4, 81 = 3^4, and 1/8 = 2^-3. For non-integer exponents, roots appear, as in a^(1/n) = the nth root of a, and fractional exponents such as a^(p/q) denote roots raised to a power p.

Exponent laws are used to manipulate potensform expressions. Key rules (for suitable bases) include: a^m * a^n = a^(m+n); (a^m)^n = a^(m n); (ab)^n = a^n b^n; a^0 = 1 for a ≠ 0; and a^-n = 1/a^n. These laws enable simplification and expansion of expressions in algebra and calculus.

The potensform is closely related to scientific notation, where large or small numbers are written as a base times a power of 10, and to prime factorization, which expresses integers as products of prime powers. It is widely used in science, engineering, and mathematics to simplify computations and to model exponential growth or decay.

Limitations arise when dealing with negative bases and non-integer exponents, where some expressions may be undefined in the real numbers. In such cases, complex numbers or restricted domains may be used. See also exponent, logarithm, scientific notation, and prime factorization.

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