 Wikipedia

# Exponentiation

"Exponent" redirects here. For other uses, see Exponent (disambiguation).

Exponentiation is a mathematical operation, written asbn, involving two numbers, the baseb and the exponent or powern, and pronounced as "b raised to the power ofn". Whenn is a positive integer, exponentiation corresponds to repeated multiplication of the base: that is,bn is the product of multiplyingn bases:

Graphs ofy = bx for various bases b: base1/2. Each curve passes through the point(0, 1) because any nonzero number raised to the power of 0 is 1. Atx = 1, the value of y equals the base because any number raised to the power of 1 is the number itself.
$b^{n}=\underbrace {b\times \dots \times b} _{n{\text{ times}}}.$

The exponent is usually shown as a superscript to the right of the base. In that case,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", or most briefly as "b to the nth".

One hasb1 = b, and, for any positive integersm andn, one hasbnbm = bn+m. To extend this property to non-positive integer exponents,b0 is defined to be1, andbn (withn a positive integer andb not zero) is defined as1/bn. In particular,b−1 is equal to1/b, the reciprocal ofb.

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.

## Contents

The term power (Latin: potentia, potestas, dignitas) 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. In The Sand Reckoner, Archimedes discovered and proved the law of exponents,10a ⋅ 10b = 10a+b, necessary to manipulate powers of10.[citation needed] 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.

In the late 16th century, Jost Bürgi used Roman numerals for exponents.

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.

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.

Some mathematicians (such as Isaac Newton) used exponents only for powers greater than two, preferring to represent squares as repeated multiplication. Thus they would write polynomials, for example, asax + bxx + cx3 + d.

Another historical synonym,[clarification needed] involution, is now rare and should not be confused with its more common meaning.

In 1748, Leonhard Euler introduced variable exponents, and, implicitly, non-integer exponents by writing:

"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."

The expressionb2 = bb is called "the square of b" or "b squared", because the area of a square with side-lengthb isb2.

Similarly, the expressionb3 = bbb is called "the cube of b" or "b cubed", because the volume of a cube with side-lengthb isb3.

When it is a positive integer, the exponent indicates how many copies of the base are multiplied together. For example,35 = 3 ⋅ 3 ⋅ 3 ⋅ 3 ⋅ 3 = 243. The base3 appears5 times in the multiplication, because the exponent is5. Here,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, so35 can be simply read "3 to the 5th", or "3 to the 5". Therefore, the exponentiationbn 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".

A formula with nested exponentiation, such as357 (which means3(57) and not(35)7), is called a tower of powers, or simply a tower.

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, and this definition can be used as soon one has an associative multiplication:

The base case is

$b^{1}=b$

and the recurrence is

$b^{n+1}=b^{n}\cdot b.$

The associativity of multiplication implies that for any positive integersm andn,

$b^{m+n}=b^{m}\cdot b^{n},$

and

$(b^{m})^{n}=b^{mn}.$

### Zero exponent

By definition, any nonzero number raised to the0 power is1:

$b^{0}=1.$

This definition is the only possible that allows extending the formula

$b^{m+n}=b^{m}\cdot b^{n}$

to zero exponents. It may be used in every algebraic structure with a multiplication that has an identity.

Intuitionally,$b^{0}$ may be interpreted as the empty product of copies ofb. So, the equality$b^{0}=1$ is a special case of the general convention for the empty product.

The case of00 is more complicated. In contexts where only integer powers are considered, the value1 is generally assigned to$0^{0},$ but, otherwise, the choice of whether to assign it a value and what value to assign may depend on context. 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 integern and nonzerob:

$b^{-n}={\frac {1}{b^{n}}}.$

Raising 0 to a negative exponent is undefined, but in some circumstances, it may be interpreted as infinity ($\infty$).

This definition of exponentiation with negative exponents is the only one that allows extending the identity$b^{m+n}=b^{m}\cdot b^{n}$ 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 denoted1 (for example, the square matrices of a given dimension). In particular, in such a structure, the inverse of an invertible elementx is standardly denoted$x^{-1}.$

### Identities and properties

"Laws of Indices" redirects here. For the horse, see Laws of Indices (horse).

The following identities, often called exponent rules, hold for all integer exponents, provided that the base is non-zero:

{\begin{aligned}b^{m+n}&=b^{m}\cdot b^{n}\\\left(b^{m}\right)^{n}&=b^{m\cdot n}\\(b\cdot c)^{n}&=b^{n}\cdot c^{n}\end{aligned}}

Unlike addition and multiplication, exponentiation is not commutative. For example,23 = 8 ≠ 32 = 9. Also unlike addition and multiplication, exponentiation is not associative. For example,(23)2 = 82 = 64, whereas2(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,

$b^{p^{q}}=b^{\left(p^{q}\right)},$

which, in general, is different from

$\left(b^{p}\right)^{q}=b^{pq}.$

### Powers of a sum

The powers of a sum can normally be computed from the powers of the summands by the binomial formula

$(a+b)^{n}=\sum _{i=0}^{n}{\binom {n}{i}}a^{i}b^{n-i}=\sum _{i=0}^{n}{\frac {n!}{i!(n-i)!}}a^{i}b^{n-i}.$

However, this formula is true only if the summands commute (i.e. thatab = ba), which is implied if they belong to a structure that is commutative. Otherwise, ifa andb 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^^ instead of^) for exponentiation with non-commuting bases, which is then called non-commutative exponentiation.

### Combinatorial interpretation

For nonnegative integersn andm, the value ofnm is the number of functions from a set ofm elements to a set ofn elements (see cardinal exponentiation). Such functions can be represented asm-tuples from ann-element set (or asm-letter words from ann-letter alphabet). Some examples for particular values ofm andn are given in the following table:

nm Thenm possiblem-tuples of elements from the set{1, ..., n}
05 = 0 none
14 = 1 (1, 1, 1, 1)
23 = 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)
32 = 9 (1, 1), (1, 2), (1, 3), (2, 1), (2, 2), (2, 3), (3, 1), (3, 2), (3, 3)
41 = 4 (1), (2), (3), (4)
50 = 1 ()

### Particular bases

#### Powers of ten

Main article: Power of 10

In the base ten (decimal) number system, integer powers of10 are written as the digit1 followed or preceded by a number of zeroes determined by the sign and magnitude of the exponent. For example,103 =1000 and10−4 =0.0001.

Exponentiation with base10 is used in scientific notation to denote large or small numbers. For instance,299792458 m/s (the speed of light in vacuum, in metres per second) can be written as2.99792458×108 m/s and then approximated as2.998×108 m/s.

SI prefixes based on powers of10 are also used to describe small or large quantities. For example, the prefix kilo means103 =1000, so a kilometre is1000 m.

#### Powers of two

Main article: Power of two

The first negative powers of2 are commonly used, and have special names, e.g.: half and quarter.

Powers of2 appear in set theory, since a set withn members has a power set, the set of all of its subsets, which has2n members.

Integer powers of2 are important in computer science. The positive integer powers2n give the number of possible values for ann-bit integer binary number; for example, a byte may take28 = 256 different values. The binary number system expresses any number as a sum of powers of2, and denotes it as a sequence of0 and1, separated by a binary point, where1 indicates a power of2 that appears in the sum; the exponent is determined by the place of this1: the nonnegative exponents are the rank of the1 on the left of the point (starting from0), and the negative exponents are determined by the rank on the right of the point.

#### Powers of one

The powers of one are all one:1n = 1.

The first power of a number is the number itself:$n^{1}=n.$

#### Powers of zero

If the exponentn is positive (n > 0), thenth power of zero is zero:0n = 0.

If the exponentn is negative (n < 0), thenth power of zero0n is undefined, because it must equal$1/0^{-n}$ withn > 0, and this would be$1/0$ according to above.

The expression00 is either defined as 1, or it is left undefined (see Zero to the power of zero).

#### Powers of negative one

Ifn is an even integer, then(−1)n = 1.

Ifn is an odd integer, then(−1)n = −1.

Because of this, powers of−1 are useful for expressing alternating sequences. For a similar discussion of powers of the complex numberi, see § Powers of complex numbers.

### Large exponents

The limit of a sequence of powers of a number greater than one diverges; in other words, the sequence grows without bound:

bn → ∞ asn → ∞ whenb > 1

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:

bn → 0 asn → ∞ when|b| < 1

Any power of one is always one:

bn = 1 for alln ifb = 1

Powers of–1 alternate between1 and–1 asn alternates between even and odd, and thus do not tend to any limit asn grows.

Ifb < –1,bn, alternates between larger and larger positive and negative numbers asn alternates between even and odd, and thus does not tend to any limit asn grows.

If the exponentiated number varies while tending to1 as the exponent tends to infinity, then the limit is not necessarily one of those above. A particularly important case is

(1 + 1/n)ne asn → ∞

See § The exponential function below.

Other limits, in particular those of expressions that take on an indeterminate form, are described in § Limits of powers below.

### Power functions

Power functions for$n=1,3,5$
Power functions for$n=2,4,6$

Real functions of the form$f(x)=cx^{n}$, where$c\neq 0$, are sometimes called power functions.[citation needed] When$n$ is an integer and$n\geq 1$, 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. Functions with this kind of symmetry($f(-x)=f(x)$) 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($f(-x)=-f(x)$) are called odd functions.

For$c<0$, the opposite asymptotic behavior is true in each case.

### Table of powers of decimal digits

n n2 n3 n4 n5 n6 n7 n8 n9 n10
1 1 1 1 1 1 1 1 1 1
2 4 8 16 32 64 128 256 512 1024
3 9 27 81 243 729 2187 6561 19683 59049
4 16 64 256 1024 4096 16384 65536 262144 1048576
5 25 125 625 3125 15625 78125 390625 1953125 9765625
6 36 216 1296 7776 46656 279936 1679616 10077696 60466176
7 49 343 2401 16807 117649 823543 5764801 40353607 282475249
8 64 512 4096 32768 262144 2097152 16777216 134217728 1073741824
9 81 729 6561 59049 531441 4782969 43046721 387420489 3486784401
10 100 1000 10000 100000 1000000 10000000 100000000 1000000000 10000000000
From top to bottom: x1/8, x1/4, x1/2, x1, x2, x4, x8.

Ifx is a nonnegative real number, andn is a positive integer,$x^{\frac {1}{n}}$ or${\sqrt[{n}]{x}}$ denotes the unique positive real nth root ofx, that is, the unique positive real numbery such that$y^{n}=x.$

Ifx is a positive real number, and${\frac {p}{q}}$ is a rational number, withp andq ≠ 0 integers, then${\textstyle x^{\frac {p}{q}}}$ is defined as

$x^{\frac {p}{q}}=\left(x^{p}\right)^{\frac {1}{q}}=(x^{\frac {1}{q}})^{p}.$

The equality on the right may be derived by setting$y=x^{\frac {1}{q}},$ and writing$(x^{\frac {1}{q}})^{p}=y^{p}=\left((y^{p})^{q}\right)^{\frac {1}{q}}=\left((y^{q})^{p}\right)^{\frac {1}{q}}=(x^{p})^{\frac {1}{q}}.$

Ifr is a positive rational number,$0^{r}=0,$ 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 realnth root, which is negative ifn is odd, and no real root ifn is even. In the latter case, whichever complexnth root one chooses for$x^{\frac {1}{n}},$ the identity$(x^{a})^{b}=x^{ab}$ cannot be satisfied. For example,

$\left((-1)^{2}\right)^{\frac {1}{2}}=1^{\frac {1}{2}}=1\neq (-1)^{2\cdot {\frac {1}{2}}}=(-1)^{1}=-1.$

See § Real exponents and § Powers of complex numbers for details on the way these problems may be handled.

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 (§ Limits of rational exponents, below), or in terms of the logarithm of the base and the exponential function (§ 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 § 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

$\left(b^{r}\right)^{s}=b^{rs}$

is true; see § 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

The limit ofe1/n ise0 = 1 whenn 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 numberb with an arbitrary real exponentx can be defined by continuity with the rule

$b^{x}=\lim _{r(\in \mathbb {Q} )\to x}b^{r}\quad (b\in \mathbb {R} ^{+},\,x\in \mathbb {R} ),$

where the limit is taken over rational values ofr only. This limit exists for every positiveb and every realx.

For example, ifx =π, the non-terminating decimal representationπ = 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$b^{\pi }:$

$\left[b^{3},b^{4}\right],\left[b^{3.1},b^{3.2}\right],\left[b^{3.14},b^{3.15}\right],\left[b^{3.141},b^{3.142}\right],\left[b^{3.1415},b^{3.1416}\right],\left[b^{3.14159},b^{3.14160}\right],\ldots$

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 positiveb and realx as a continuous function ofb andx. See also Well-defined expression.

### The exponential function

Main article: Exponential function

The exponential function is often defined as$x\mapsto e^{x},$ where$e\approx 2.718$ is Euler's number. For avoiding 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

$\exp(x)=e^{x}.$

There are many equivalent ways to define the exponential function, one of them being

$\exp(x)=\lim _{n\rightarrow \infty }\left(1+{\frac {x}{n}}\right)^{n}.$

One has$\exp(0)=1,$ and the exponential identity$\exp(x+y)=\exp(x)\exp(y)$ holds as well, since

$\exp(x)\exp(y)=\lim _{n\rightarrow \infty }\left(1+{\frac {x}{n}}\right)^{n}\left(1+{\frac {y}{n}}\right)^{n}=\lim _{n\rightarrow \infty }\left(1+{\frac {x+y}{n}}+{\frac {xy}{n^{2}}}\right)^{n},$

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$\exp(x)=e^{x}$ whenx is an integer (this results from the repeated-multiplication definition of the exponentiation). Ifx is real,$\exp(x)=e^{x}$ results from the definitions given in preceding sections, by using the exponential identity ifx is rational, and the continuity of the exponential function otherwise.

The limit that defines the exponential function converges for every complex value ofx, and therefore it can be used to extend the definition of$\exp(z)$, and thus$e^{z},$ from the real numbers to any complex argumentz. This extended exponential function still satifies the exponential identity, and is commonly used for defining exponentiation for complex base and exponent.

### Powers via logarithms

The definition ofex as the exponential function allows definingbx for every positive real numbersb, in terms of exponential and logarithm function. Specifically, the fact that the natural logarithmln(x) is the inverse of the exponential functionex means that one has

$b=\exp(\ln b)=e^{\ln b}$

for everyb > 0. For preserving the identity$(e^{x})^{y}=e^{xy},$ one must have

$b^{x}=\left(e^{\ln b}\right)^{x}=e^{x\ln b}$

So,$e^{x\ln b}$ can be used as an alternative definition ofbx for any positive realb. This agrees with the definition given above using rational exponents and continuity, with the advantage to extend straightforwardly to any complex exponent.

Ifb is a positive real number, exponentiation with baseb and complex exponent is defined by mean of the exponential function with complex argument (see the end of § The exponential function, above) as

$b^{z}=e^{(z\ln b)},$

where$\ln b$ denotes the natural logarithm ofb.

This satisfies the identity

$b^{z+t}=b^{z}b^{t},$

In general,${\textstyle \left(b^{z}\right)^{t}}$ is not defined, sincebz is not a real number. If a meaning is given to the exponentiation of a complex number (see § Powers of complex numbers, below), one has, in general,

$\left(b^{z}\right)^{t}\neq b^{zt},$

unlessz is real orw is integer.

$e^{iy}=\cos y+i\sin y,$

allows expressing the polar form of$b^{z}$ in terms of the real and imaginary parts ofz, namely

$b^{x+iy}=b^{x}(\cos(y\ln b)+i\sin(y\ln b)),$

where the absolute value of the trigonometric factor is one. This results from

$b^{x+iy}=b^{x}b^{iy}=b^{x}e^{iy\ln b}=b^{x}(\cos(y\ln b)+i\sin(y\ln b)).$

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 ofnth roots, that is, of exponents$1/n,$ wheren is a positive integer. Although the general theory of exponentiation with non-integer exponents applies tonth roots, this case deserves to be considered first, since it does not need to use complex logarithms, and is therefore easier to understand.

### nth roots of a complex number

Every nonzero complex numberz may be written in polar form as

$z=\rho e^{i\theta }=r(cos\theta +i\sin \theta ),$

where$\rho$ is the absolute value ofz, and$\theta$ is its argument. The argument is defined up to an integer multiple of2π; this means that, if$\theta$ is the argument of a complex number, then$\theta +2k\pi$ is also an argument of the same complex number.

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 annth root of a complex number can be obtained by taking thenth root of the absolute value and dividing its argument byn:

$\left(\rho e^{i\theta }\right)^{\frac {1}{n}}={\sqrt[{n}]{\rho }}\,e^{\frac {i\theta }{n}}.$

If$2i\pi$ is added to$\theta ,$ the complex number in not changed, but this adds$2i\pi /n$ to the argument of thenth root, and provides a newnth root. This can be donen times, and provides thennth roots of the complex number.

It is usual to choose one of thennth root as the principal root. The common choice is to choose thenth root for which$-\pi <\theta \leq \pi ,$ that is, thenth root that has the largest real part, and, if they are two, the one with positive imaginary part. This makes the principalnth root a continuous function in the whole complex plane, except for negative real values of the radicand. This function equals the usualnth root for positive real radicands. For negative real radicands, and odd exponents, the principalnth root is not real, although the usualnth root is real. Analytic continuation shows that the principalnth root is the unique complex differentiable function that extends the usualnth root to the complex plane without the nonpositive real numbers.

If the complex number is moved around zero by increasing its argument, after an increment of$2\pi ,$ the complex number comes back to its initial position, and itsnth roots are permuted circularly (they are multiplied by$e^{2i\pi /n}$). This shows that it is not possible to define anth root function that is not continuous in the whole complex plane.

#### Roots of unity

Main article: Root of unity
The three third roots of 1

Thenth roots of unity are then complex numbers such thatwn = 1, wheren is a positive integer. They arise in various areas of mathematics, such as in discrete Fourier transform or algebraic solutions of algebraic equations (Lagrange resolvent).

Thennth roots of unity are then first powers of$\omega =e^{\frac {2\pi i}{n}}$, that is$1=\omega ^{0}=\omega ^{n},\omega =\omega ^{1},\omega ^{2},\omega ^{n-1}.$ Thenth roots of unity that have this generating property are called primitiventh roots of unity; they have the form$\omega ^{k}=e^{\frac {2k\pi i}{n}},$ withk coprime withn. The unique primitive square root of unity is$-1;$ the primitive fourth roots of unity are$i$ and$-i.$

Thenth roots of unity allow expressing allnth roots of a complex numberz as then products of a givennth roots ofz with anth root of unity.

Geometrically, thenth roots of unity lie on the unit circle of the complex plane at the vertices of a regularn-gon with one vertex on the real number 1.

As the number$e^{\frac {2k\pi i}{n}}$ is the primitiventh root of unity with the smallest positive argument, it is called the principal primitiventh root of unity, sometimes shortened as principalnth root of unity, although this terminology can be confused with the principal value of$1^{1/n}$ which is 1.

### Complex exponentiation

Defining exponentiation with complex bases leads to difficulties that are similar to those described in the preceding section, except that there are, in general, infinitely many possible values for$z^{w}$. So, either a principal value is defined, which is not continuous for the values ofz that are real and nonpositive, or$z^{w}$ is defined as a multivalued function.

In all cases, the complex logarithm is used to define complex exponentiation as

$z^{w}=e^{w\log z},$

where$\log z$ is the variant of the complex logarithm that is used, which is, a function or a multivalued function such that

$e^{\log z}=z$

for everyz in its domain of definition.

#### Principal value

The principal value of the complex logarithm is the unique function, commonly denoted$\log ,$ such that, for every nonzero complex numberz,

$e^{\log z}=z,$

and the imaginary part ofz satisfies

$-\pi <\mathrm {Im} \leq \pi .$

The principal value of the complex logarithm is not defined for$z=0,$ it is discontinuous at negative real values ofz, and it is holomorphic (that is, complex differentiable) elsewhere. Ifz is real and positive, the principal value of the complex logarithm is the natural logarithm:$\log z=\ln z.$

The principal value of$z^{w}$ is defined as$z^{w}=e^{w\log z},$ where$\log z$ is the principal value of the logarithm.

The function$(z,w)\to z^{w}$ is holomorphic except in the neibourhood of the points wherez is real and nonpositive.

Ifz is real and positive, the principal value of$z^{w}$ equals its usual value defined above. If$w=1/n,$ wheren is an integer, this principal value is the same as the one defined above.

#### Multivalued function

In some contexts, there is a problem with the discontinuity of the principal values of$\log z$ and$z^{w}$ at the negative real values ofz. In this case, it is useful to consider these functions as multivalued functions.

If$\log z$ denotes one of the values of the multivalued logarithm (typically its principal value), the other values are$2ik\pi +\log z,$ wherek is any integer. Similarly, if$z^{w}$ is one value of the exponentiation, then the other values are given by

$e^{w(2ik\pi +\log z)}=z^{w}e^{2ik\pi w},$

wherek is any integer.

Different values ofk give different values of$z^{w}$ unlessw is a rational number, that is, there is an integerd such thatdw is an integer. This results from the periodicity of the exponential function, more specifically, that$e^{a}=e^{b}$ if and only if$a-b$ is an integer multiple of$2\pi i.$

If$w={\frac {m}{n}}$ is a rational number withm andn coprime integers with$n>0,$ then$z^{w}$ has exactlyn values. In the case$m=1,$ these values are the same as those described in §nth roots of a complex number. Ifw is an integer, there is only one value that agrees with that of § Integer exponents.

The multivalued exponentiation is holomorphic for$z\neq 0,$ in the sense that its graph consists of several sheets that define each a holomorphic function in the neighborhood of every point. Ifz varies continuously along a circle around0, then, after a turn, the value of$z^{w}$ has changed of sheet.

#### Computation

The canonical form$x+iy$ of$z^{w}$ can be computed from the canonical form ofz andw. Although this can be described by a single formula, it is clearer to split the computation in several steps.

• Polar form ofz. If$z=a+ib$ is the canonical form ofz (a andb being real), then its polar form is
$z=\rho e^{i\theta }=\rho (\cos \theta +i\sin \theta ),$
where$\rho ={\sqrt {a^{2}+b^{2}}}$ and$\theta =\operatorname {atan2} (a,b)$ (see atan2 for the definition of this function).
• Logarithm ofz. The principal value of this logarithm is$\log z=\ln \rho +i\theta ,$ where$\ln$ denotes the natural logarithm. The other values of the logarithm are obtained by adding$2ik\pi$ for any integerk.
• Canonical form of$w\log z.$ If$w=c+di$ withc andd real, the values of$w\log z$ are
$w\log z=(c\ln \rho -d\theta -2dk\pi )+i(d\ln \rho +c\theta +2ck\pi ),$
the principal value corresponding to$k=0.$
• Final result. Using the identities$e^{x+y}e^{x}=e^{y}$ and$e^{y\ln x}=x^{y},$ one gets
$z^{w}=\rho ^{c}e^{-d(\theta +2k\pi )}\left(\cos(d\ln \rho +c\theta +2ck\pi )+i\sin(d\ln \rho +c\theta +2ck\pi )\right),$
with$k=0$ for the principal value.
##### Examples
• $i^{i}$
The polar form ofi is$i=e^{i\pi /2},$ and the values of$\log i$ are thus
$\log i=i\left({\frac {\pi }{2}}+2k\pi \right).$
It follows that
$i^{i}=e^{i\log i}=e^{-{\frac {\pi }{2}}}e^{-2k\pi }.$
So, all values of$i^{i}$ are real, the principal one being
$e^{-{\frac {\pi }{2}}}\approx 0.2079.$
• $(-2)^{3+4i}$
Similarly, the polar form of−2 is$-2=2e^{i\pi }.$ So, the above described method gives the values
{\begin{aligned}(-2)^{3+4i}&=2^{3}e^{-4(\pi +2k\pi )}(\cos(4\ln 2+3(\pi +2k\pi ))+i\sin(4\ln 2+3(\pi +2k\pi )))\\&=-2^{3}e^{-4(\pi +2k\pi )}(\cos(4\ln 2)+i\sin(4\ln 2)).\end{aligned}}
In this case, all the values have the same argument$4\ln 2,$ and different absolute values.

In both examples, all values of$z^{w}$ have the same argument. More generally, this is true if and only if the real part ofw is an integer.

#### Failure of power and logarithm identities

Some identities for powers and logarithms for positive real numbers will fail for complex numbers, no matter how complex powers and complex logarithms are defined as single-valued functions. For example:

• The identitylog(bx) = x ⋅ log b holds wheneverb is a positive real number andx is a real number. But for the principal branch of the complex logarithm one has
$\log((-i)^{2})=\log(-1)=i\pi \neq 2\log(-i)=2\log(e^{-i\pi /2})=2\,{\frac {-i\pi }{2}}=-i\pi$

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:

$\log w^{z}\equiv z\log w{\pmod {2\pi i}}$

This identity does not hold even when considering log as a multivalued function. The possible values oflog(wz) contain those ofz ⋅ log w as a proper subset. UsingLog(w) for the principal value oflog(w) andm,n as any integers the possible values of both sides are:

{\begin{aligned}\left\{\log w^{z}\right\}&=\left\{z\cdot \operatorname {Log} w+z\cdot 2\pi in+2\pi im\mid m,n\in \mathbb {Z} \right\}\\\left\{z\log w\right\}&=\left\{z\operatorname {Log} w+z\cdot 2\pi in\mid n\in \mathbb {Z} \right\}\end{aligned}}
• The identities(bc)x = bxcx and(b/c)x = bx/cx are valid whenb andc are positive real numbers andx is a real number. But, for the principal values, one has
$(-1\cdot -1)^{\frac {1}{2}}=1\not =(-1)^{\frac {1}{2}}(-1)^{\frac {1}{2}}=-1$

and

$\left({\frac {1}{-1}}\right)^{\frac {1}{2}}=(-1)^{\frac {1}{2}}=i\not ={\frac {1^{\frac {1}{2}}}{(-1)^{\frac {1}{2}}}}={\frac {1}{i}}=-i$

On the other hand, whenx is an integer, the identities are valid for all nonzero complex numbers.

If exponentiation is considered as a multivalued function then the possible values of(−1 ⋅ −1)1/2 are{1, −1}. The identity holds, but saying{1} = {(−1 ⋅ −1)1/2} is wrong.
• The identity(ex)y = exy holds for real numbersx andy, but assuming its truth for complex numbers leads to the following paradox, discovered in 1827 by Clausen: For any integern, we have:
1. $e^{1+2\pi in}=e^{1}e^{2\pi in}=e\cdot 1=e$
2. $\left(e^{1+2\pi in}\right)^{1+2\pi in}=e\qquad$ (taking the$(1+2\pi in)$-th power of both sides)
3. $e^{1+4\pi in-4\pi ^{2}n^{2}}=e\qquad$ (using$\left(e^{x}\right)^{y}=e^{xy}$ and expanding the exponent)
4. $e^{1}e^{4\pi in}e^{-4\pi ^{2}n^{2}}=e\qquad$ (using$e^{x+y}=e^{x}e^{y}$)
5. $e^{-4\pi ^{2}n^{2}}=1\qquad$ (dividing bye)
but this is false when the integern is nonzero. The error is the following: by definition,$e^{y}$ is a notation for$\exp(y),$ a true function, and$x^{y}$ is a notation for$\exp(y\log x),$ which is a multi-valued function. Thus the notation is ambiguous whenx = e. Here, before expanding the exponent, the second line should be
$\exp \left((1+2\pi in)\log \exp(1+2\pi in)\right)=\exp(1+2\pi in).$
Therefore, when expanding the exponent, one has implicitly supposed that$\log \exp z=z$ for complex values ofz, which is wrong, as the complex logarithm is multivalued. In other words, the wrong identity(ex)y = exy must be replaced by the identity
$\left(e^{x}\right)^{y}=e^{y\log e^{x}},$
which is a true identity between multivalued functions.

Ifb is a positive real algebraic number, andx is a rational number, thenbx is an algebraic number. This results from the theory of algebraic extensions. This remains true ifb is any algebraic number, in which case, all values ofbx (as a multivalued function) are algebraic. Ifx is irrational (that is, not rational), and bothb andx are algebraic, Gelfond–Schneider theorem asserts that all values ofbx are transcendental (that is, not algebraic), except ifb equals0 or1.

In other words, ifx is irrational and$b\not \in \{0,1\},$ then at least one ofb,x andbx is transcendental.

The definition of exponentiation with positive integer exponents as repeated multiplication may apply to any associative operation denoted as a multiplication. The definition of$x^{0}$ requires further the existence of a multiplicative identity.

An algebraic structure consisting of a set together with an associative operation denoted multiplicatively, and a multiplicative identity denoted by 1 is a monoid. In such a monoid, exponentiation of an elementx is defined inductively by

• $x^{0}=1,$
• $x^{n+1}=xx^{n}$ for every nonnegative integern.

Ifn is a negative integer,$x^{n}$ is defined only ifx has a multiplicative inverse. In this case, the inverse ofx is denoted$x^{-1},$ and$x^{n}$ is defined as$\left(x^{-1}\right)^{-n}.$

Exponentiation with integer exponents obeys the following laws, forx andy in the algebraic structure, andm andn integers:

{\begin{aligned}x^{0}&=1\\x^{m+n}&=x^{m}x^{n}\\(x^{m})^{n}&=x^{mn}\\(xy)^{n}&=x^{n}y^{n}\quad {\text{if }}xy=yx,{\text{and, in particular, if the multiplication is commutative.}}\end{aligned}}

These definitions are widely used in many areas of mathematics, notably for groups, rings, fields, square matrices (which form a ring). They apply also to functions from a set to itself, which form a monoid under function composition. This includes, as specific instances, geometric transformations, and endomorphisms of any mathematical structure.

When there are several operations that may be repeated, it is common to indicate the repeated operation by placing its symbol in the superscript, before the exponent. For example, iff is a real function whose valued can be multiplied,$f^{n}$ denotes the exponentiation with respect of multiplication, and$f^{\circ n}$ may denote exponentiation with respect of function composition. That is,

$(f^{n})(x)=(f(x))^{n}=f(x)\,f(x)\cdots f(x),$

and

$(f^{\circ n})(x)=f(f(\cdots f(f(x))\cdots )).$

Commonly,$(f^{n})(x)$ is denoted$f(x)^{n},$ while$(f^{\circ n})(x)$ is denoted$f^{n}(x).$

### In a group

A multiplicative group is a set with as associative operation denoted as multiplication, that has an identity element, and such that every element has an inverse.

So, ifG is a group,$x^{n}$ is defined for every$x\in G$ and every integern.

The set of all powers of an element of a group form a subgroup. A group (or subgroup) that consists of all powers of a specific elementx is the cyclic group generated byx. If all the powers ofx are distinct, the group is isomorphic to the additive group$\mathbb {Z}$ of the integers. Otherwise, the cyclic group is finite (it has a finite number of elements), and its number of elements is the order ofx. If the order ofx isn, then$x^{n}=x^{0}=1,$ and the cyclic group generated byx consists of then first powers ofx (starting indifferently from the exponent0 or1).

Order of elements play a fundamental role in group theory. For example, the order of an element in a finite group is always a divisor of the number of elements of the group (the order of the group). The possible orders of group elements are important in the study of the structure of a group (see Sylow theorems), and in the classification of finite simple groups.

Superscript notation is also used for conjugation; that is,gh = h−1gh, where g and h are elements of a group. This notation cannot be confused with exponentiation, since the superscript is not an integer. The motivation of this notation is that conjugation obeys some of the laws of exponentiation, namely$(g^{h})^{k}=g^{hk}$ and$(gh)^{k}=g^{k}h^{k}.$

### In a ring

In a ring, it may occurs that some nonzero elements satisfy$x^{n}=0$ for some integern. Such an element is said nilpotent. In a commutative ring, the nilpotent elements form an ideal, called the nilradical of the ring.

If the nilradical is reduced to the zero ideal (that is, if$x\neq 0$ implies$x^{n}\neq 0$ for every positive integern), the commutative ring is said reduced. Reduced rings important in algebraic geometry, since the coordinate ring of an affine algebraic set is always a reduced ring.

More generally, given an idealI in a commutative ringR, the set of the elements ofR that have a power inI is an ideal, called the radical ofI. The nilradical is the radical of the zero ideal. A radical ideal is an ideal that equals its own radical. In a polynomial ring$k[x_{1},\ldots ,x_{n}]$ over a fieldk, an ideal is radical if and only if it is the set of all polynomials that are zero on an affine algebraic set (this is a consequence of Hilbert's Nullstellensatz).

### Matrices and linear operators

If A is a square matrix, then the product of A with itself n times is called the matrix power. Also$A^{0}$ is defined to be the identity matrix, and if A is invertible, then$A^{-n}=\left(A^{-1}\right)^{n}$.

Matrix powers appear often in the context of discrete dynamical systems, where the matrix A expresses a transition from a state vector x of some system to the next state Ax of the system. This is the standard interpretation of a Markov chain, for example. Then$A^{2}x$ is the state of the system after two time steps, and so forth:$A^{n}x$ is the state of the system after n time steps. The matrix power$A^{n}$ is the transition matrix between the state now and the state at a time n steps in the future. So computing matrix powers is equivalent to solving the evolution of the dynamical system. In many cases, matrix powers can be expediently computed by using eigenvalues and eigenvectors.

Apart from matrices, more general linear operators can also be exponentiated. An example is the derivative operator of calculus,$d/dx$, which is a linear operator acting on functions$f(x)$ to give a new function$(d/dx)f(x)=f'(x)$. The n-th power of the differentiation operator is the n-th derivative:

$\left({\frac {d}{dx}}\right)^{n}f(x)={\frac {d^{n}}{dx^{n}}}f(x)=f^{(n)}(x).$

These examples are for discrete exponents of linear operators, but in many circumstances it is also desirable to define powers of such operators with continuous exponents. This is the starting point of the mathematical theory of semigroups. Just as computing matrix powers with discrete exponents solves discrete dynamical systems, so does computing matrix powers with continuous exponents solve systems with continuous dynamics. Examples include approaches to solving the heat equation, Schrödinger equation, wave equation, and other partial differential equations including a time evolution. The special case of exponentiating the derivative operator to a non-integer power is called the fractional derivative which, together with the fractional integral, is one of the basic operations of the fractional calculus.

### Finite fields

Main article: Finite field

A field is an algebraic structure in which multiplication, addition, subtraction, and division are defined and satisfy the properties that multiplication is associative and every nonzero element has a multiplicative inverse. This implies that exponentiation with integer exponents is well-defined, except for nonpositive powers of0. Common examples are the complex numbers and their subfields, the rational numbers and the real numbers, which have been considered earlier in this article, and are all infinite.

A finite field is a field with a finite number of elements. This number of elements is either a prime number or a prime power; that is, it has the form$q=p^{k},$ wherep is a prime number, andk is a positive integer. For every suchq, there are fields withq elements. The fields withq elements are all isomorphic, which allows, in general, working as if there were only one field withq elements, denoted$\mathbb {F} _{q}.$

One has

$x^{q}=x$

for every$x\in \mathbb {F} _{q}.$

A primitive element in$\mathbb {F} _{q}$ is an elementg such the set of theq − 1 first powers ofg (that is,$\{g^{1}=g,g^{2},\ldots ,g^{p-1}=g^{0}=1\}$) equals the set of the nonzero elements of$\mathbb {F} _{q}.$ There are$\varphi (p-1)$ primitive elements in$\mathbb {F} _{q},$ where$\varphi$ is Euler's totient function.

In$\mathbb {F} _{q},$ the Freshman's dream identity

$(x+y)^{p}=x^{p}+y^{p}$

is true for the exponentp. As$x^{p}=x$ in$\mathbb {F} _{q},$ It follows that the map

{\begin{aligned}F\colon {}&\mathbb {F} _{q}\to \mathbb {F} _{q}\\&x\mapsto x^{p}\end{aligned}}

is linear over$\mathbb {F} _{q},$ and is a field automorphism, called the Frobenius automorphism. If$q=p^{k},$ the field$\mathbb {F} _{q}$ hask automorphisms, which are thek first powers (under composition) ofF. In other words, the Galois group of$\mathbb {F} _{q}$ is cyclic of orderk, generated by the Frobenius automorphism.

The Diffie–Hellman key exchange is an application of exponentiation in finite fields that is widely used for secure communications. It uses the fact that exponentiation is computationally inexpensive, whereas the inverse operation, the discrete logarithm, is computationally expensive. More precisely, ifg is a primitive element in$\mathbb {F} _{q},$ then$g^{e}$ can be efficiently computed with exponentiation by squaring for anye, even ifq is large, while there is no known algorithm allowing retrievinge from$g^{e}$ ifq is sufficiently large.

The Cartesian product of two setsS andT is the set of the ordered pairs$(x,y)$ such that$x\in S$ and$y\in T.$ This operation is not properly commutative nor associative, but has these properties up to canonical isomorphisms, that allow identifying, for example,$(x,(y,z)),$$((x,y),z),$ and$(x,y,z).$

This allows defining thenth power$S^{n}$ of a setS as the set of alln-tuples$(x_{1},\ldots ,x_{n})$ of elements ofS.

WhenS is endowed with some structure, it is frequent that$S^{n}$ is naturally endowed with a similar structure. In this case, the term "direct product" is generally used instead of "Cartesian product", and exponentiation denotes product structure. For example$\mathbb {R} ^{n}$ (where$\mathbb {R}$ denotes the real numbers) denotes the Cartesian product ofn copies of$\mathbb {R} ,$ as well as their direct product as vector space, topological spaces, rings, etc.

### Sets as exponents

An-tuple$(x_{1},\ldots ,x_{n})$ of elements ofS can be considered as a function from$\{1,\ldots ,n\}.$ This generalizes to the following notation.

Given two setsS andT, the set of all functions fromT toS is denoted$S^{T}$ This exponential notation is justified by the following canonical isomorphisms (for the first one, see Currying):

$(S^{T})^{U}\cong S^{T\times U},$
$S^{T\sqcup U}\cong S^{T}\times S^{U},$

where$\times$ denotes the Cartesian product, and$\sqcup$ the disjoint union.

One can use sets as exponents for other operations on sets, typically for direct sums of abelian groups, vector spaces, or modules. For distinguishing direct sums from direct products, the exponent of a direct sum is placed between parentheses. For example,$\mathbb {R} ^{\mathbb {N} }$ denotes the vector space of the infinite sequences of real numbers, and$\mathbb {R} ^{(\mathbb {N} )}$ the vector space of those sequences that have a finite number of nonzero elements. The latter has a basis consisting of the sequences with exactly one nonzero element that equals1, while the Hamel bases of the former cannot be explicitly described (because there existence involves Zorn's lemma).

In this context,2 can represents the set$\{0,1\}.$ So,$2^{S}$ denotes the power set ofS, that is the set of the functions fromS to$\{0,1\},$ which can be identified with the set of the subsets ofS, by mapping each function to the inverse image of1.

This fits in with the exponentiation of cardinal numbers, in the sense that|ST| = |S||T|, where|X| is the cardinality ofX.

### In category theory

In the category of sets, the morphisms between setsX andY are the functions fromX toY. It results that the set of the functions fromX toY that is denoted$Y^{X}$ in the preceding section can also be denoted$\hom(X,Y).$ The isomorphism$(S^{T})^{U}\cong S^{T\times U}$ can be rewritten

$\hom(U,S^{T})\cong \hom(T\times U,S).$

This means the functor "exponentiation to the powerT" is a right adjoint to the functor "direct product withT".

This generalizes to the definition of exponentiation in a category in which finite direct products exist: in such a category, the functor$X\to X^{T}$ is, if it exists, a right adjoint to the functor$Y\to T\times Y.$ A category is called a Cartesian closed category, if direct products exist, and the functor$Y\to X\times Y$ has a right adjoint for everyT.

Main articles: Tetration and Hyperoperation

Just as exponentiation of natural numbers is motivated by repeated multiplication, it is possible to define an operation based on repeated exponentiation; this operation is sometimes called hyper-4 or tetration. Iterating tetration leads to another operation, and so on, a concept named hyperoperation. This sequence of operations is expressed by the Ackermann function and Knuth's up-arrow notation. Just as exponentiation grows faster than multiplication, which is faster-growing than addition, tetration is faster-growing than exponentiation. Evaluated at(3, 3), the functions addition, multiplication, exponentiation, and tetration yield 6, 9, 27, and7625597484987 (= 327 = 333 = 33) respectively.

Zero to the power of zero gives a number of examples of limits that are of the indeterminate form 00. The limits in these examples exist, but have different values, showing that the two-variable functionxy has no limit at the point(0, 0). One may consider at what points this function does have a limit.

More precisely, consider the functionf(x, y) = xy defined onD = {(x, y) ∈ R2 : x > 0}. ThenD can be viewed as a subset ofR2 (that is, the set of all pairs(x, y) withx,y belonging to the extended real number lineR = [−∞, +∞], endowed with the product topology), which will contain the points at which the functionf has a limit.

In fact,f has a limit at all accumulation points ofD, except for(0, 0),(+∞, 0),(1, +∞) and(1, −∞). Accordingly, this allows one to define the powersxy by continuity whenever0 ≤ x ≤ +∞,−∞ ≤ y ≤ +∞, except for 00, (+∞)0, 1+∞ and 1−∞, which remain indeterminate forms.

Under this definition by continuity, we obtain:

• x+∞ = +∞ andx−∞ = 0, when1 < x ≤ +∞.
• x+∞ = 0 andx−∞ = +∞, when0 ≤ x < 1.
• 0y = 0 and(+∞)y = +∞, when0 < y ≤ +∞.
• 0y = +∞ and(+∞)y = 0, when−∞ ≤ y < 0.

These powers are obtained by taking limits ofxy for positive values ofx. This method does not permit a definition ofxy whenx < 0, since pairs(x, y) withx < 0 are not accumulation points ofD.

On the other hand, whenn is an integer, the powerxn is already meaningful for all values ofx, including negative ones. This may make the definition0n = +∞ obtained above for negativen problematic whenn is odd, since in this casexn → +∞ asx tends to0 through positive values, but not negative ones.

Computing bn using iterated multiplication requiresn − 1 multiplication operations, but it can be computed more efficiently than that, as illustrated by the following example. To compute 2100, apply Horner's rule to the exponent 100 written in binary:

$100=2^{2}+2^{5}+2^{6}=2^{2}(1+2^{3}(1+2))$.

Then compute the following terms in order, reading Horner's rule from right to left.

 22 = 4 2 (22) = 23 = 8 (23)2 = 26 = 64 (26)2 = 212 =4096 (212)2 = 224 =16777216 2 (224) = 225 =33554432 (225)2 = 250 =1125899906842624 (250)2 = 2100 =1267650600228229401496703205376

This series of steps only requires 8 multiplications instead of 99.

In general, the number of multiplication operations required to computebn can be reduced to$\sharp n+\lfloor \log _{2}n\rfloor -1,$ by using exponentiation by squaring, where$\sharp n$ denotes the number of1 in the binary representation ofn. For some exponents (100 is not among them), the number of multiplications can be further reduced by computing and using the minimal addition-chain exponentiation. Finding the minimal sequence of multiplications (the minimal-length addition chain for the exponent) forbn is a difficult problem, for which no efficient algorithms are currently known (see Subset sum problem), but many reasonably efficient heuristic algorithms are available. However, in practical computations, exponentiation by squaring is efficient enough, and much more easy to implement.

Function composition is a binary operation that is defined on functions such that the codomain of the function written on the right is included in the domain of the function written on the left. It is denoted$g\circ f,$ and defined as

$(g\circ f)(x)=g(f(x))$