In mathematics, exponentiation is an operation involving two numbers, the base and the exponent or power. Exponentiation is written as , where is the base and is the power; this is pronounced as " (raised) to the (power of) ". When is a positive integer, exponentiation corresponds to repeated multiplication of the base: that is, is the product of multiplying bases:
The exponent is usually shown as a superscript to the right of the base. In that case, is called "b raised to the nth power", "b (raised) to the power of n", "the nth power of b", "b to the nth power", or most briefly as "b to the nth".
Starting from the basic fact stated above that, for any positive integer , is occurrences of all multiplied by each other, several other properties of exponentiation directly follow. In particular:
In other words, when multiplying a base raised to one exponent by the same base raised to another exponent, the exponents add. From this basic rule that exponents add, we can derive that must be equal to 1 for any , as follows. For any , . Dividing both sides by gives .
The fact that can similarly be derived from the same rule. For example, . Taking the cube root of both sides gives .
The rule that multiplying makes exponents add can also be used to derive the properties of negative integer exponents. Consider the question of what should mean. In order to respect the "exponents add" rule, it must be the case that . Dividing both sides by gives , which can be more simply written as , using the result from above that . By a similar argument, .
The properties of fractional exponents also follow from the same rule. For example, suppose we consider and ask if there is some suitable exponent, which we may call , such that . From the definition of the square root, we have that . Therefore, the exponent must be such that . Using the fact that multiplying makes exponents add gives . The on the right-hand side can also be written as , giving . Equating the exponents on both sides, we have . Therefore, , so .
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.
Etymology
The term exponent originates from the Latin exponentem, the present participle of exponere, meaning "to put forth". The term power () is a mistranslation of the ancient Greek δύναμις (dúnamis, here: "amplification") used by the Greek mathematician Euclid for the square of a line, following Hippocrates of Chios.
History
Antiquity
The Sand Reckoner
In The Sand Reckoner, Archimedes proved the law of exponents, , necessary to manipulate powers of . He then used powers of to estimate the number of grains of sand that can be contained in the universe.
Islamic Golden Age
Māl and kaʿbah ("square" and "cube") In the 9th century, the Persian mathematician Muhammad ibn Mūsā al-Khwārizmī used the terms مَال (māl, "possessions", "property") for a square—the Muslims, "like most mathematicians of those and earlier times, thought of a squared number as a depiction of an area, especially of land, hence property"—and كَعْبَة (kaʿbah, "cube") for a cube, which later Islamic mathematicians represented in mathematical notation as the letters mīm (m) and kāf (k), respectively, by the 15th century, as seen in the work of Abū al-Hasan ibn Alī al-Qalasādī.
15th–18th century
Introducing exponents
Nicolas Chuquet used a form of exponential notation in the 15th century, for example to represent . This was later used by Henricus Grammateus and Michael Stifel in the 16th century. In the late 16th century, Jost Bürgi would use Roman numerals for exponents in a way similar to that of Chuquet, for example for .
"Exponent"; "square" and "cube" The word exponent was coined in 1544 by Michael Stifel. 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.