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| name = notation| variant1 caption = base b and exponent n}}File:Expo02.svg|thumb|315px|Graphs of {{math|1=y = bx}} for various bases {{math|b}}:{{nobr|{{legend-line|inline=yes|green solid 2px|base {{math|10}},}}}}{{nobr|{{legend-line|inline=yes|red solid 2px|base {{math|e}},}}}}{{nobr|{{legend-line|inline=yes|blue solid 2px|base {{math|2}}base {{math|2}}{{Arithmetic operations}}In mathematics, exponentiation is an operation involving two numbers: the base and the exponent or power. Exponentiation is written as {{math|b’n}}, where {{mvar|b}} is the base and {{mvar|n}} is the power; this is pronounced as “{{mvar|b}} (raised) to the (power of) {{mvar|n}}”.WEB, Nykamp, Duane, Basic rules for exponentiation, Math Insight,mathinsight.org/exponentiation_basic_rules, August 27, 2020, When {{mvar|n}} is a positive integer, exponentiation corresponds to repeated multiplication of the base: that is, {{math|b’n}} is the product of multiplying {{mvar|n}} bases:b^n = underbrace{b times b times dots times b times b}_{n text{ times}}.The exponent is usually shown as a superscript to the right of the base. In that case, {{math|bn}} 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”,{{MathWorld |title=Power |id=Power |access-date=2020-08-27}} or most briefly as ”b to the n(th)”.Starting from the basic fact stated above that, for any positive integer n, b^n is n occurrences of b all multiplied by each other, several other properties of exponentiation directly follow. In particular:There are three common notations for multiplication: xtimes y is most commonly used for explicit numbers and at a very elementary level; xy is most common when variables are used; xcdot y is used for emphasizing that one talks of multiplication or when omitting the multiplication sign would be confusing.begin{align}b^{n+m} & = underbrace{b times dots times b}_{n+m text{ times}} [1ex]& = underbrace{b times dots times b}_{n text{ times}} times underbrace{b times dots times b}_{m text{ times}} [1ex]& = b^n times b^mend{align}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 b^0 must be equal to 1 for any b neq 0, as follows. For any n, b^0 times b^n = b^{0+n} = b^n. Dividing both sides by b^n gives b^0 = b^n / b^n = 1.The fact that b^1 = b can similarly be derived from the same rule. For example, (b^1)^3 = b^1 times b^1 times b^1 = b^{1+1+1} = b^3 . Taking the cube root of both sides gives b^1 = b.The rule that multiplying makes exponents add can also be used to derive the properties of negative integer exponents. Consider the question of what b^{-1} should mean. In order to respect the “exponents add” rule, it must be the case that b^{-1} times b^1 = b^{-1+1} = b^0 = 1 . Dividing both sides by b^{1} gives b^{-1} = 1 / b^1, which can be more simply written as b^{-1} = 1 / b, using the result from above that b^1 = b. By a similar argument, b^{-n} = 1 / b^n.The properties of fractional exponents also follow from the same rule. For example, suppose we consider sqrt{b} and ask if there is some suitable exponent, which we may call r, such that b^r = sqrt{b}. From the definition of the square root, we have that sqrt{b} times sqrt{b} = b . Therefore, the exponent r must be such that b^r times b^r = b . Using the fact that multiplying makes exponents add gives b^{r+r} = b . The b on the right-hand side can also be written as b^1 , giving b^{r+r} = b^1 . Equating the exponents on both sides, we have r+r = 1 . Therefore, r = frac{1}{2} , so sqrt{b} = b^{1/2} .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”.WEB, Exponent | Etymology of exponent by etymonline,www.etymonline.com/word/exponent, The term power () is a mistranslationBOOK, Rotman, Joseph J., Joseph J. Rotman, 2015, Advanced Modern Algebra, Part 1, Providence, RI, American Mathematical Society, p. 130, fn. 4, 978-1-4704-1554-9, 3rd, Graduate Studies in Mathematics, 165,www.ams.org/books/gsm/165/04, subscription, BOOK, Szabó, Árpád, 1978, The Beginnings of Greek Mathematics, Dordrecht, D. Reidel, 37, 90-277-0819-3, Synthese Historical Library, 17, A.M. Ungar,archive.org/details/TheBeginningsOfGreekMathematics, of the ancient Greek δύναμις (dúnamis, here: “amplification“) used by the Greek mathematician Euclid for the square of a line, following Hippocrates of Chios.BOOK, Ball, W. W. Rouse, W. W. Rouse Ball, 1915, A Short Account of the History of Mathematics, London, Macmillan Publishers, Macmillan, 38, 6th,archive.org/details/shortaccountofhi00ballrich,

History

Antiquity

The Sand Reckoner

In The Sand Reckoner, Archimedes proved the law of exponents, {{math|1=10a · 10b = 10a+b}}, necessary to manipulate powers of {{math|10}}.Archimedes. (2009). THE SAND-RECKONER. In T. Heath (Ed.), The Works of Archimedes: Edited in Modern Notation with Introductory Chapters (Cambridge Library Collection - Mathematics, pp. 229-232). Cambridge: Cambridge University Press. {{doi|10.1017/CBO9780511695124.017}}. He then used powers of {{math|10}} to estimate the number of grains of sand that can be contained in the universe.

Islamic Golden Age

Māl and kaÊ¿bah (“square” and “cube“)“>

Māl and kaÊ¿bah (“square” and “cube“)

In the 9th century, the Persian mathematician Al-Khwarizmi 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 (KaÊ¿bah, “cube“) for a cube, which later Islamic mathematicians represented in mathematical notation as the letters mÄ«m (m) and kāf (k), respectively, by the 15th century, as seen in the work of Abu’l-Hasan ibn Ali al-Qalasadi.{{MacTutor|id=Al-Qalasadi|title= Abu’l Hasan ibn Ali al Qalasadi}}

15th–18th century

Introducing exponents

Nicolas Chuquet used a form of exponential notation in the 15th century, for example {{math|122}} to represent {{math|12x2}}.BOOK, Cajori, Florian, A History of Mathematical Notations, 1928, The Open Court Company, 1, 102,archive.org/details/historyofmathema031756mbp, 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 {{Overset|iii|4}} for {{math|4x3}}.BOOK, Cajori, Florian, Florian Cajori, 1928, A History of Mathematical Notations, London, Open Court Publishing Company, 344, 1,archive.org/details/historyofmathema031756mbp,

“Exponent”; “square” and “cube”

The word exponent was coined in 1544 by Michael Stifel.WEB,jeff560.tripod.com/e.html, Earliest Known Uses of Some of the Words of Mathematics (E), June 23, 2017, BOOK, Stifel, Michael, Michael Stifel, 1544, Arithmetica integra, Nuremberg, Johannes Petreius, 235v,archive.org/details/bub_gb_fndPsRv08R0C/page/n491, In the 16th century, Robert Recorde used the terms square, cube, zenzizenzic (fourth power), sursolid (fifth), zenzicube (sixth), second sursolid (seventh), and zenzizenzizenzic (eighth).WEB, Zenzizenzizenzic, World Wide Words, Quinion, Michael, Michael Quinion,www.worldwidewords.org/weirdwords/ww-zen1.htm, 2020-04-16, Biquadrate has been used to refer to the fourth power as well.

Modern exponential notation

In 1636, James Hume used in essence modern notation, when in L’algèbre de Vietè he wrote {{math|Aiii}} for {{math|A3}}.BOOK, Cajori, Florian, A History of Mathematical Notations, 1928, The Open Court Company, 1, 204,archive.org/details/historyofmathema031756mbp, Early in the 17th century, the first form of our modern exponential notation was introduced by René Descartes in his text titled La Géométrie; there, the notation is introduced in Book I.BOOK, Descartes, René, René Descartes, 1637, Discourse de la méthode [...], Leiden, Jan Maire, 299, La Géométrie,gallica.bnf.fr/ark:/12148/btv1b86069594/f383.image, Et aa, ou {{math, a2, , pour multiplier {{math|a}} par soy mesme; Et {{math|a3}}, pour le multiplier encore une fois par {{math|a}}, & ainsi a l’infini}} (And {{math|aa}}, or {{math|a2}}, in order to multiply {{math|a}} by itself; and {{math|a3}}, in order to multiply it once more by {{math|a}}, and thus to infinity).{{Blockquote|text=I designate ... {{math|aa}}, or {{math|a2}} in multiplying {{math|a}} by itself; and {{math|a3}} in multiplying it once more again by {{math|a}}, and thus to infinity.|author=René Descartes|title=La Géométrie}}Some mathematicians (such as Descartes) used exponents only for powers greater than two, preferring to represent squares as repeated multiplication. Thus they would write polynomials, for example, as {{math|ax + bxx + cx3 + d}}.

“Indices”

Samuel Jeake introduced the term indices in 1696.{{MacTutor|class=Miscellaneous|id=Mathematical_notation|title=Etymology of some common mathematical terms}} The term involution was used synonymously with the term indices, but had declined in usageThe most recent usage in this sense cited by the OED is from 1806 (OED, involution, ). and should not be confused with its more common meaning.

Variable exponents, non-integer exponents

In 1748, Leonhard Euler introduced variable exponents, and, implicitly, non-integer exponents by writing:{{blockquote|Consider exponentials or powers in which the exponent itself is a variable. It is clear that quantities of this kind are not algebraic functions, since in those the exponents must be constant.}}

Terminology

The expression {{math|1=b2 = b · b}} is called “the square of {{math|b}}” or “{{math|b}} squared”, because the area of a square with side-length {{math|b}} is {{math|b2}}. (It is true that it could also be called “{{math|b}} to the second power”, but “the square of {{math|b}}” and “{{math|b}} squared” are so ingrained by tradition and convenience that “{{math|b}} to the second power” tends to sound unusual or clumsy.)Similarly, the expression {{math|1=b3 = b · b · b}} is called “the cube of {{math|b}}” or “{{math|b}} cubed”, because the volume of a cube with side-length {{math|b}} is {{math|b3}}.When an exponent is a positive integer, that exponent indicates how many copies of the base are multiplied together. For example, {{math|1=35 = 3 · 3 · 3 · 3 · 3 = 243}}. The base {{math|3}} appears {{math|5}} times in the multiplication, because the exponent is {{math|5}}. Here, {{math|243}} is the 5th power of 3, or 3 raised to the 5th power.The word “raised” is usually omitted, and sometimes “power” as well, so {{math|35}} can be simply read “3 to the 5th”, or “3 to the 5”. Therefore, the exponentiation {{math|bn}} can be expressed as ”b to the power of n”, ”b to the nth power”, ”b to the nth”, or most briefly as ”b to the n”.

Integer exponents

The exponentiation operation with integer exponents may be defined directly from elementary arithmetic operations.

Positive exponents

The definition of the exponentiation as an iterated multiplication can be formalized by using induction,BOOK, Abstract Algebra: an inquiry based approach, Hodge, Jonathan K., Schlicker, Steven, Sundstorm, Ted, 94, 2014, CRC Press, 978-1-4665-6706-1,books.google.com/books?id=qToTAgAAQBAJ&pg=PA94, and this definition can be used as soon one has an associative multiplication:The base case is and the recurrence is The associativity of multiplication implies that for any positive integers {{mvar|m}} and {{mvar|n}}, and

Zero exponent

As mentioned earlier, a (nonzero) number raised to the {{math|0}} power is {{math|1}}:BOOK, Technical Shop Mathematics, Achatz, Thomas, 101, 2005, 3rd, Industrial Press, 978-0-8311-3086-2,books.google.com/books?id=YOdtemSmzQQC&pg=PA101, This value is also obtained by the empty product convention, which may be used in every algebraic structure with a multiplication that has an identity. This way the formula also holds for n=0.The case of {{math|00}} is controversial. In contexts where only integer powers are considered, the value {{math|1}} is generally assigned to {{math|00}} but, otherwise, the choice of whether to assign it a value and what value to assign may depend on context. {{Crossreference|For more details, see Zero to the power of zero.}}

Negative exponents

Exponentiation with negative exponents is defined by the following identity, which holds for any integer {{mvar|n}} and nonzero {{mvar|b}}: Raising 0 to a negative exponent is undefined but, in some circumstances, it may be interpreted as infinity (infty).BOOK This definition of exponentiation with negative exponents is the only one that allows extending the identity b^{m+n}=b^mcdot b^n to negative exponents (consider the case m=-n).The same definition applies to invertible elements in a multiplicative monoid, that is, an algebraic structure, with an associative multiplication and a multiplicative identity denoted {{math|1}} (for example, the square matrices of a given dimension). In particular, in such a structure, the inverse of an invertible element {{mvar|x}} is standardly denoted x^{-1}.

Identities and properties

{{Redirect|Laws of Indices|the horse|Laws of Indices (horse)}}The following identities, often called {{vanchor|exponent rules}}, hold for all integer exponents, provided that the base is non-zero: end{align}Unlike addition and multiplication, exponentiation is not commutative. For example, {{math|1=23 = 8 ≠ 32 = 9}}. Also unlike addition and multiplication, exponentiation is not associative. For example, {{math|1=(23)2 = 82 = 64}}, whereas {{math|1=2(32) = 29 = 512}}. Without parentheses, the conventional order of operations for serial exponentiation in superscript notation is top-down (or right-associative), not bottom-up (or left-associative). That is, which, in general, is different from

Powers of a sum

The powers of a sum can normally be computed from the powers of the summands by the binomial formula However, this formula is true only if the summands commute (i.e. that {{math|1=ab = ba}}), which is implied if they belong to a structure that is commutative. Otherwise, if {{mvar|a}} and {{mvar|b}} are, say, square matrices of the same size, this formula cannot be used. It follows that in computer algebra, many algorithms involving integer exponents must be changed when the exponentiation bases do not commute. Some general purpose computer algebra systems use a different notation (sometimes {{math|^^}} instead of {{math|^}}) for exponentiation with non-commuting bases, which is then called non-commutative exponentiation.

Combinatorial interpretation

{{See also|#Exponentiation over sets|l1=Exponentiation over sets}}For nonnegative integers {{mvar|n}} and {{mvar|m}}, the value of {{math|nm}} is the number of functions from a set of {{mvar|m}} elements to a set of {{mvar|n}} elements (see cardinal exponentiation). Such functions can be represented as {{mvar|m}}-tuples from an {{mvar|n}}-element set (or as {{mvar|m}}-letter words from an {{mvar|n}}-letter alphabet). Some examples for particular values of {{mvar|m}} and {{mvar|n}} are given in the following table:{| class=“wikitable“!{{math|nm}}!The {{math|nm}} possible {{mvar|m}}-tuples of elements from the set {{math|{{mset|1, ..., n}}}}5}} = 0none}}4}} = 1|(1, 1, 1, 1)3}} = 8|(1, 1, 1), (1, 1, 2), (1, 2, 1), (1, 2, 2), (2, 1, 1), (2, 1, 2), (2, 2, 1), (2, 2, 2)2}} = 9|(1, 1), (1, 2), (1, 3), (2, 1), (2, 2), (2, 3), (3, 1), (3, 2), (3, 3)1}} = 4|(1), (2), (3), (4)0}} = 1|()

Particular bases

Powers of ten

{{See also|Scientific notation}}In the base ten (decimal) number system, integer powers of {{math|10}} are written as the digit {{math|1}} followed or preceded by a number of zeroes determined by the sign and magnitude of the exponent. For example, {{math|1={{val|e=3}} = {{val|1000}}}} and {{math|1={{val|e=-4}} = {{val|0.0001}}}}.Exponentiation with base {{math|10}} is used in scientific notation to denote large or small numbers. For instance, {{val|299792458|u=m/s}} (the speed of light in vacuum, in metres per second) can be written as {{val|2.99792458|e=8|u=m/s}} and then approximated as {{val|2.998|e=8|u=m/s}}.SI prefixes based on powers of {{math|10}} are also used to describe small or large quantities. For example, the prefix kilo means {{math|1={{val|e=3}} = {{val|1000}}}}, so a kilometre is {{val|1000|u=metres}}.

{{anchor|Base 2}}Powers of two

The first negative powers of {{math|2}} are commonly used, and have special names, e.g.: half and quarter.Powers of {{math|2}} appear in set theory, since a set with {{math|n}} members has a power set, the set of all of its subsets, which has {{math|2n}} members.Integer powers of {{math|2}} are important in computer science. The positive integer powers {{math|2n}} give the number of possible values for an {{math|n}}-bit integer binary number; for example, a byte may take {{math|1=28 = 256}} different values. The binary number system expresses any number as a sum of powers of {{math|2}}, and denotes it as a sequence of {{math|0}} and {{math|1}}, separated by a binary point, where {{math|1}} indicates a power of {{math|2}} that appears in the sum; the exponent is determined by the place of this {{math|1}}: the nonnegative exponents are the rank of the {{math|1}} on the left of the point (starting from {{math|0}}), and the negative exponents are determined by the rank on the right of the point.

Powers of one

Every power of one equals: {{math|1=1n = 1}}. This is true even if {{mvar|n}} is negative.The first power of a number is the number itself: {{math|1=n1 = n}}.

Powers of zero

If the exponent {{mvar|n}} is positive ({{math|n > 0}}), the {{mvar|n}}th power of zero is zero: {{math|1=0n = 0}}.If the exponent {{mvar|n}} is negative ({{math|n < 0}}), the {{mvar|n}}th power of zero {{math|0n}} is undefined, because it must equal 1/0^{-n} with {{math|−n > 0}}, and this would be 1/0 according to above.The expression {{math|00}} is either defined as {{math|1}}, or it is left undefined.

Powers of negative one

If {{math|n}} is an even integer, then {{math|1=(−1)n = 1}}. This is because a negative number multiplied by another negative number cancels the sign, and thus gives a positive number.If {{math|n}} is an odd integer, then {{math|1=(−1)n = −1}}. This is because there will be a remaining {{math|−1}} after removing {{math|−1}} pairs.Because of this, powers of {{math|−1}} are useful for expressing alternating sequences. For a similar discussion of powers of the complex number {{math|i}}, see {{slink||nth roots of a complex number}}.

Large exponents

The limit of a sequence of powers of a number greater than one diverges; in other words, the sequence grows without bound: This can be read as ”b to the power of n tends to +∞ as n tends to infinity when b is greater than one”.Powers of a number with absolute value less than one tend to zero: Any power of one is always one: Powers of {{math|–1}} alternate between {{math|1}} and {{math|–1}} as {{math|n}} alternates between even and odd, and thus do not tend to any limit as {{math|n}} grows.If {{math|b < –1}}, {{math|1=bn}} alternates between larger and larger positive and negative numbers as {{math|n}} alternates between even and odd, and thus does not tend to any limit as {{math|n}} grows.If the exponentiated number varies while tending to {{math|1}} as the exponent tends to infinity, then the limit is not necessarily one of those above. A particularly important case is See {{slink||Exponential function}} below.Other limits, in particular those of expressions that take on an indeterminate form, are described in {{slink||Limits of powers}} below.

Power functions

(File:Potenssi 1 3 5.svg|thumb|left|Power functions for {{math|1=n = 1, 3, 5}})(File:Potenssi 2 4 6.svg|thumb|Power functions for {{math|1=n = 2, 4, 6}})Real functions of the form f(x) = cx^n, where c ne 0, are sometimes called power functions.BOOK, Hass, Joel R., Heil, Christopher E., Weir, Maurice D., Thomas, George B., Thomas’ Calculus, 2018, Pearson, 9780134439020, 7–8, 14, When n is an integer and n ge 1, two primary families exist: for n even, and for n odd. In general for c > 0, when n is even f(x) = cx^n will tend towards positive infinity with increasing x, and also towards positive infinity with decreasing x. All graphs from the family of even power functions have the general shape of y=cx^2, flattening more in the middle as n increases.BOOK, Anton, Howard, Bivens, Irl, Davis, Stephen, Calculus: Early Transcendentals, 2012, John Wiley & Sons, 28, 9780470647691, 9th,archive.org/details/calculusearlytra00anto_656, limited, Functions with this kind of symmetry {{nobr|(f(-x)= f(x))}} are called even functions.When n is odd, f(x)’s asymptotic behavior reverses from positive x to negative x. For c > 0, f(x) = cx^n will also tend towards positive infinity with increasing x, but towards negative infinity with decreasing x. All graphs from the family of odd power functions have the general shape of y=cx^3, flattening more in the middle as n increases and losing all flatness there in the straight line for n=1. Functions with this kind of symmetry {{nobr|(f(-x)= -f(x))}} are called odd functions.For c < 0, the opposite asymptotic behavior is true in each case.

Table of powers of decimal digits{|class“wikitable” style@text-align:right”

! n !! n2 !! n3 !! n4 !! n5 !! n6 !! n7 !! n8 !! n9 !! n101 >| 12 >| 10243 >2,187}} {{val19,683}} {{val|59,049}}4 >4,096}} {{val65,536}} {{val1,048,576}}5 >15,625}} {{val390,625}} {{val9,765,625}}6 >7,776}} {{val279,936}} {{val10,077,696}} {{val|60,466,176}}7 >16,807}} {{val823,543}} {{val40,353,607}} {{val|282,475,249}}8 >32,768}} {{val2,097,152}} {{val134,217,728}} {{val|1,073,741,824}}9 >59,049}} {{val4,782,969}} {{val387,420,489}} {{val|3,486,784,401}}10 >10,000}} {{val1,000,000}} {{val100,000,000}} {{val10,000,000,000}}

Rational exponents

(File:Mplwp roots 01.svg|right|thumb|300px|From top to bottom: {{math|x1/8}}, {{math|x1/4}}, {{math|x1/2}}, {{math|x1}}, {{math|x2}}, {{math|x4}}, {{math|x8}}.)If {{mvar|x}} is a nonnegative real number, and {{mvar|n}} is a positive integer, x^{1/n} or sqrt[n]x denotes the unique positive real {{mvar|n}}th root of {{mvar|x}}, that is, the unique positive real number {{mvar|y}} such that y^n=x.If {{mvar|x}} is a positive real number, and frac pq is a rational number, with {{mvar|p}} and {{mvar|q > 0}} integers, then x^{p/q} is defined as The equality on the right may be derived by setting y=x^frac 1q, and writing (x^frac 1q)^p=y^p=left((y^p)^qright)^frac 1q=left((y^q)^pright)^frac 1q=(x^p)^frac 1q.If {{mvar|r}} is a positive rational number, {{math|1=0r = 0}}, by definition.All these definitions are required for extending the identity (x^r)^s = x^{rs} to rational exponents.On the other hand, there are problems with the extension of these definitions to bases that are not positive real numbers. For example, a negative real number has a real {{mvar|n}}th root, which is negative, if {{mvar|n}} is odd, and no real root if {{mvar|n}} is even. In the latter case, whichever complex {{mvar|n}}th root one chooses for x^frac 1n, the identity (x^a)^b=x^{ab} cannot be satisfied. For example, See {{slink||Real exponents}} and {{slink||Non-integer powers of complex numbers}} for details on the way these problems may be handled.

Real exponents

For positive real numbers, exponentiation to real powers can be defined in two equivalent ways, either by extending the rational powers to reals by continuity ({{slink||Limits of rational exponents}}, below), or in terms of the logarithm of the base and the exponential function ({{slink||Powers via logarithms}}, below). The result is always a positive real number, and the identities and properties shown above for integer exponents remain true with these definitions for real exponents. The second definition is more commonly used, since it generalizes straightforwardly to complex exponents.On the other hand, exponentiation to a real power of a negative real number is much more difficult to define consistently, as it may be non-real and have several values (see {{slink||Real exponents with negative bases}}). One may choose one of these values, called the principal value, but there is no choice of the principal value for which the identity is true; see {{slink||Failure of power and logarithm identities}}. Therefore, exponentiation with a basis that is not a positive real number is generally viewed as a multivalued function.

Limits of rational exponents

(File:Continuity of the Exponential at 0.svg|thumb|The limit of {{math|e{{sup|1/n}}}} is {{math|1=e{{sup|0}} = 1}} when {{mvar|n}} tends to the infinity.)Since any irrational number can be expressed as the limit of a sequence of rational numbers, exponentiation of a positive real number {{mvar|b}} with an arbitrary real exponent {{mvar|x}} can be defined by continuity with the ruleBOOK, Elements of Real Analysis, Denlinger, Charles G., Jones and Bartlett, 2011, 278–283, 978-0-7637-7947-4, where the limit is taken over rational values of {{mvar|r}} only. This limit exists for every positive {{mvar|b}} and every real {{mvar|x}}.For example, if {{math|1=x = {{pi}}}}, the non-terminating decimal representation {{math|1=π = 3.14159...}} and the monotonicity of the rational powers can be used to obtain intervals bounded by rational powers that are as small as desired, and must contain b^pi: So, the upper bounds and the lower bounds of the intervals form two sequences that have the same limit, denoted b^pi.This defines b^x for every positive {{mvar|b}} and real {{mvar|x}} as a continuous function of {{mvar|b}} and {{mvar|x}}. See also Well-defined expression.BOOK, Limits of sequences, {{Google books, ecTsDAAAQBAJ, 154, yes, |title=Analysis I |series=Texts and Readings in Mathematics |year=2016 |last1=Tao |first1=Terence |volume=37 |pages=126–154 |isbn=978-981-10-1789-6 |doi=10.1007/978-981-10-1789-6_6}}

Exponential function

The exponential function is often defined as xmapsto e^x, where eapprox 2.718 is Euler’s number. To avoid circular reasoning, this definition cannot be used here. So, a definition of the exponential function, denoted exp(x), and of Euler’s number are given, which rely only on exponentiation with positive integer exponents. Then a proof is sketched that, if one uses the definition of exponentiation given in preceding sections, one has There are many equivalent ways to define the exponential function, one of them being One has exp(0)=1, and the exponential identity exp(x+y)=exp(x)exp(y) holds as well, since and the second-order term frac{xy}{n^2} does not affect the limit, yielding exp(x)exp(y) = exp(x+y).Euler’s number can be defined as e=exp(1). It follows from the preceding equations that exp(x)=e^x when {{mvar|x}} is an integer (this results from the repeated-multiplication definition of the exponentiation). If {{mvar|x}} is real, exp(x)=e^x results from the definitions given in preceding sections, by using the exponential identity if {{mvar|x}} is rational, and the continuity of the exponential function otherwise.The limit that defines the exponential function converges for every complex value of {{mvar|x}}, and therefore it can be used to extend the definition of exp(z), and thus e^z, from the real numbers to any complex argument {{mvar|z}}. This extended exponential function still satisfies the exponential identity, and is commonly used for defining exponentiation for complex base and exponent.

Powers via logarithms

The definition of {{math|e’x}} as the exponential function allows defining {{math|b’x}} for every positive real numbers {{mvar|b}}, in terms of exponential and logarithm function. Specifically, the fact that the natural logarithm {{math|ln(x)}} is the inverse of the exponential function {{math|ex}} means that one has for every {{math|b > 0}}. For preserving the identity (e^x)^y=e^{xy}, one must have So, e^{x ln b} can be used as an alternative definition of {{math|bx}} for any positive real {{mvar|b}}. This agrees with the definition given above using rational exponents and continuity, with the advantage to extend straightforwardly to any complex exponent.

Complex exponents with a positive real base

If {{mvar|b}} is a positive real number, exponentiation with base {{mvar|b}} and complex exponent {{mvar|z}} is defined by means of the exponential function with complex argument (see the end of {{slink||Exponential function}}, above) as where ln b denotes the natural logarithm of {{mvar|b}}.This satisfies the identity In general,left(b^zright)^t is not defined, since {{math|bz}} is not a real number. If a meaning is given to the exponentiation of a complex number (see {{slink||Non-integer powers of complex numbers}}, below), one has, in general, unless {{mvar|z}} is real or {{mvar|t}} is an integer.Euler’s formula, allows expressing the polar form of b^z in terms of the real and imaginary parts of {{mvar|z}}, namely where the absolute value of the trigonometric factor is one. This results from

Non-integer powers of complex numbers

In the preceding sections, exponentiation with non-integer exponents has been defined for positive real bases only. For other bases, difficulties appear already with the apparently simple case of {{mvar|n}}th roots, that is, of exponents 1/n, where {{mvar|n}} is a positive integer. Although the general theory of exponentiation with non-integer exponents applies to {{mvar|n}}th roots, this case deserves to be considered first, since it does not need to use complex logarithms, and is therefore easier to understand.

{{mvar|n}}th roots of a complex number

Every nonzero complex number {{mvar|z}} may be written in polar form as where rho is the absolute value of {{mvar|z}}, and theta is its argument. The argument is defined up to an integer multiple of {{math|2{{pi}}}}; this means that, if theta is the argument of a complex number, then theta +2kpi is also an argument of the same complex number for every integer k.The polar form of the product of two complex numbers is obtained by multiplying the absolute values and adding the arguments. It follows that the polar form of an {{mvar|n}}th root of a complex number can be obtained by taking the {{mvar|n}}th root of the absolute value and dividing its argument by {{mvar|n}}: If 2pi is added to theta, the complex number is not changed, but this adds 2ipi/n to the argument of the {{mvar|n}}th root, and provides a new {{mvar|n}}th root. This can be done {{mvar|n}} times, and provides the {{mvar|n}} {{mvar|n}}th roots of the complex number.It is usual to choose one of the {{mvar|n}} {{mvar|n}}th root as the principal root. The common choice is to choose the {{mvar|n}}th root for which -pi