poweridentity

Poweridentity is a term used to describe a class of algebraic identities that govern how exponentiation interacts with multiplication, division, and other powers. These identities, often called exponent rules or power laws, are fundamental tools for simplifying expressions and solving equations involving powers.

Common power identities include: (ab)^n = a^n b^n for integer n; (a^m)^n = a^{mn} for bases where the power is defined; a^0 = 1 for a ≠ 0; a^{-n} = 1/a^n; (a/b)^n = a^n/b^n for nonzero b; and a^{m+n} = a^m a^n. These rules enable straightforward manipulation of expressions such as turning (2x^3)^4 into 16 x^{12} or (xy)^3 into x^3 y^3.

Domain considerations are important. For real exponents, the identities are most straightforward when the base a is positive. When exponents are integers, the identities hold for all nonzero bases, while with non-integer exponents and negative bases, definitions may require extending to complex numbers or choosing principal values.

Applications of power identities span algebra, calculus, and beyond. They simplify polynomials and rational expressions, convert between logarithmic and exponential forms, and aid in solving exponential equations or rearranging expressions in physics and engineering.

Limitations and caveats include issues with zero bases and negative bases under non-integer exponents, which can lead to undefined or multi-valued expressions in the real number system. In such cases, careful attention to the domain and the chosen definitions is required.

See also: exponent rules, logarithms, algebraic identities.