Exponents

An exponent expresses repeated multiplication. If a is a number and n is a nonnegative integer, a^n means a multiplied by itself n times; for example, 3^4 = 81. The base a is typically a real number, and the exponent n is often a nonnegative integer in basic arithmetic. When a is nonzero, the case n = 0 gives a^0 = 1. The value 0^0 is indeterminate in most contexts.

For negative exponents: a^{-n} = 1 / a^n, provided a ≠ 0. This expresses the reciprocal of a^n.

Exponent rules: a^m a^n = a^{m+n} (product rule), a^m / a^n = a^{m-n} (quotient rule) for a ≠ 0; (a^m)^n = a^{mn} (power rule); (ab)^n = a^n b^n.

Rational exponents and roots: For integers p and q > 0, a^{p/q} can be defined as the q-th root of a^p, provided a ≥ 0 if q is even; more generally over complex numbers one can define a^{p/q} using principal roots.

Examples: 2^3 = 8; 2^{-3} = 1/8; (3^4)^2 = 3^8; (2×5)^3 = 2^3 × 5^3 = 1000.

Applications: Exponents are central to exponential growth and decay, compound interest, and many areas of algebra and calculus. The inverse operation to exponentiation is the logarithm.