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Exponentiation is a mathematical operation, written as ''b''''n'', involving two numbers, the base ''b'' and the exponent ''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: : In that case, ''bn'' is called the ''n''-th power of ''b'', or ''b'' raised to the power ''n''. The exponent is usually shown as a superscript to the right of the base. Some common exponents have their own names: the exponent 2 (or 2nd power) is called the ''square'' of ''b'' (''b''2) or ''b squared''; the exponent 3 (or 3rd power) is called the ''cube'' of ''b'' (''b''3) or ''b cubed''. The exponent −1 of ''b'', or , is called the ''reciprocal'' of ''b''. When ''n'' is a negative integer and ''b'' is not zero, ''b''''n'' is naturally defined as 1/''b''−''n'', preserving the property . 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. == History of the notation == 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.〔For further analysis see The Sand Reckoner.〕 In the 9th century, the Persian mathematician Muhammad ibn Mūsā al-Khwārizmī used the terms ''mal'' for a square and ''kab'' 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.〔Cajori, Florian (2007). A History of Mathematical Notations; Vol I. Cosimo Classics. Pg 344 ISBN 1602066841〕 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.〔René Descartes, ''Discourse de la Méthode'' ... (Leiden, (Netherlands): Jan Maire, 1637), appended book: ''La Géométrie'', book one, (page 299. ) From page 299: ''" ... Et ''aa'', ou ''a''2, pour multiplier ''a'' par soy mesme; Et ''a''3, pour le multiplier encore une fois par ''a'', & ainsi a l'infini ; ... "'' ( ... and ''aa'', or ''a''2, in order to multiply ''a'' by itself; and ''a''3, in order to multiply it once more by ''a'', and thus to infinity ; ... )〕 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.〔See: * (Earliest Known Uses of Some of the Words of Mathematics ) * Michael Stifel, ''Arithmetica integra'' (Nuremberg ("Norimberga"), (Germany): Johannes Petreius, 1544), Liber III (Book 3), Caput III (Chapter 3): De Algorithmo numerorum Cossicorum. (On algorithms of algebra.), (page 236. ) Stifel was trying to conveniently represent the terms of geometric progressions. He devised a cumbersome notation for doing that. On page 236, he presented the notation for the first eight terms of a geometric progression (using 1 as a base) and then he wrote: ''"Quemadmodum autem hic vides, quemlibet terminum progressionis cossicæ, suum habere exponentem in suo ordine (ut 1ze habet 1. 1ʓ habet 2 &c.) sic quilibet numerus cossicus, servat exponentem suæ denominationis implicite, qui ei serviat & utilis sit, potissimus in multiplicatione & divisione, ut paulo inferius dicam."'' (However, you see how each term of the progression has its exponent in its order (as 1ze has a 1, 1ʓ has a 2, etc.), so each number is implicitly subject to the exponent of its denomination, which (turn ) is subject to it and is useful mainly in multiplication and division, as I will mention just below.) ( Most of Stifel's cumbersome symbols were taken from Christoff Rudolff, who in turn took them from Leonardo Fibonacci's ''Liber Abaci'' (1202), where they served as shorthand symbols for the Latin words ''res''/''radix'' (x), ''census''/''zensus'' (''x''2), and ''cubus'' (''x''3). )〕 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. Some mathematicians (e.g., Isaac Newton) used exponents only for powers greater than two, preferring to represent squares as repeated multiplication. Thus they would write polynomials, for example, as ''ax'' + ''bxx'' + ''cx''3 + ''d''. Another historical synonym, involution,〔This definition of "involution" appears in the OED second edition, 1989, and Merriam-Webster online dictionary (). The most recent usage in this sense cited by the OED is from 1806.〕 is now rare and should not be confused with its more common meaning. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Exponentiation」の詳細全文を読む スポンサード リンク
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