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", or "the n-th power of b".
When n is a positive integer and b is not zero, b−n is naturally defined as 1/, preserving the property bn × bm = bn + m. With exponent −1, b−1 is equal to 1/, and is 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.
The term power was used by the Greek mathematician Euclid for the square of a line. Archimedes discovered and proved the law of exponents, 10a 10b = 10a+b, necessary to manipulate powers of 10. In the 9th century, the Persian mathematician Muhammad ibn Mūsā al-Khwārizmī used the terms mal for a square and kahb for a cube, which later Islamic mathematicians represented in mathematical notation as m and k, respectively, by the 15th century, as seen in the work of Abū al-Hasan ibn Alī al-Qalasādī.