Exponent

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:which, in general, is different from If b is an algebraic number different from 0 and 1, and x an irrational algebraic number, then all values of bx (there are infinitely many) are transcendental (that is, not algebraic).Regardless of which branch of the logarithm is used, a similar failure of the identity will exist. The best that can be said (if only using this result) is that:and 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: 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. The term power was used by the Greek mathematician Euclid for the square of a line, following Hippocrates of Chios. 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. In the late 16th century, Jost Bürgi used Roman numerals for exponents. Early in the 17th century, the first form of our modern exponential notation was introduced by Rene Descartes in his text titled La Géométrie; there, the notation is introduced in Book I. Nicolas Chuquet used a form of exponential notation in the 15th century, which was later used by Henricus Grammateus and Michael Stifel in the 16th century. The word 'exponent' was coined in 1544 by Michael Stifel. Samuel Jeake introduced the term indices in 1696. In the 16th century Robert Recorde used the terms square, cube, zenzizenzic (fourth power), sursolid (fifth), zenzicube (sixth), second sursolid (seventh), and zenzizenzizenzic (eighth). Biquadrate has been used to refer to the fourth power as well.