Exponentiation is a mathematical operation, written as bn, involving two numbers, the base b and the exponent or power n. When n is a positive integer, exponentiation corresponds to repeated multiplication of the base: that is, bn is the product of multiplying n bases: b n = b × ⋯ × b ⏟ n times . {\displaystyle b^{n}=\underbrace {b\times \dots \times b} _{n\,{\textrm {times}}}.} The exponent is usually shown as a superscript to the right of the base. In that case, bn is called "b raised to the n-th power", "b raised to the power of n", "the n-th power of b", "b to the nth", or most briefly as "b to the n". For any positive integers m and n, one has bn ⋅ bm = bn+m. To extend this property to non-positive integer exponents, b0 is defined to be 1, and b−n 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.


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