Exponentiation

Exponentiation is a mathematical operation, written as bn, involving two numbers, the base b and the exponent or power n, and pronounced as "b raised to the power of n".[1][2] When n is a positive integer, exponentiation corresponds to repeated multiplication of the base: that is, bn is the product of multiplying n bases:[2]

Graphs of y = bx for various bases b:   base 10,   base e,   base 2,   base 1/2. Each curve passes through the point (0, 1) because any nonzero number raised to the power of 0 is 1. At x = 1, the value of y equals the base because any number raised to the power of 1 is the number itself.

The exponent is usually shown as a superscript to the right of the base. In that case, bn is called "b raised to the nth power", "b raised to the power of n",[1] "the nth power of b", "b to the nth power",[3] or most briefly as "b to the nth".

One has b1 = b, and, for any positive integers m and n, one has bnbm = bn+m. To extend this property to non-positive integer exponents, b0 is defined to be 1, and bn (with n a positive integer and b not zero) is defined as 1/bn. In particular, b−1 is equal to 1/b, the reciprocal of b.

The definition of exponentiation can be extended to allow any real or complex exponent. Exponentiation by integer exponents can also be defined for a wide variety of algebraic structures, including matrices.

Exponentiation is used extensively in many fields, including economics, biology, chemistry, physics, and computer science, with applications such as compound interest, population growth, chemical reaction kinetics, wave behavior, and public-key cryptography.