Exponentiation

In mathematics, exponentiation, denoted bⁿ, is an operation involving two numbers: the base, b, and the exponent or power, n. When n is a positive integer, exponentiation corresponds to repeated multiplication of the base: that is, bⁿ is the product of multiplying n bases:$${\displaystyle b^{n}=\underbrace {b\times b\times \dots \times b\times b} _{n{\text{ times}}}.}$$In particular, ${\displaystyle b^{1}=b}$.

Graphs of y = b^x for various bases b:   base 10,   base e,   base 2,   base ⁠1/2⁠. Each curve passes through the point (0, 1) because any nonzero number raised to the power of 0 is 1. At x = 1, the value of y equals the base because any number raised to the power of 1 is the number itself.

The exponent is usually shown as a superscript to the right of the base as bⁿ or in computer code as b^n. This binary operation is often read as "b to the power n"; it may also be called "b raised to the nth power", "the nth power of b", or most briefly "b to the n".

The above definition of ${\displaystyle b^{n}}$ immediately implies several properties, in particular the multiplication rule:

$${\displaystyle {\begin{aligned}b^{n}\times b^{m}&=\underbrace {b\times \dots \times b} _{n{\text{ times}}}\times \underbrace {b\times \dots \times b} _{m{\text{ times}}}\\[1ex]&=\underbrace {b\times \dots \times b} _{n+m{\text{ times}}}\ =\ b^{n+m}.\end{aligned}}}$$

That is, when multiplying a base raised to one power times the same base raised to another power, the powers add. Extending this rule to the power zero gives ${\displaystyle b^{0}\times b^{n}=b^{0+n}=b^{n}}$, and dividing both sides by ${\displaystyle b^{n}}$ gives ${\displaystyle b^{0}=b^{n}/b^{n}=1}$. That is, the multiplication rule implies the definition $${\displaystyle b^{0}=1.}$$A similar argument implies the definition for negative integer powers: $${\displaystyle b^{-n}=1/b^{n}.}$$That is, extending the multiplication rule gives ${\displaystyle b^{-n}\times b^{n}=b^{-n+n}=b^{0}=1}$. Dividing both sides by ${\displaystyle b^{n}}$ gives ${\displaystyle b^{-n}=1/b^{n}}$. This also implies the definition for fractional powers: $${\displaystyle b^{n/m}={\sqrt[{m}]{b^{n}}}.}$$For example, ${\displaystyle b^{1/2}\times b^{1/2}=b^{1/2\,+\,1/2}=b^{1}=b}$, meaning ${\displaystyle (b^{1/2})^{2}=b}$, which is the definition of square root: ${\displaystyle b^{1/2}={\sqrt {b}}}$.

The definition of exponentiation can be extended in a natural way (preserving the multiplication rule) to define ${\displaystyle b^{x}}$ for any positive real base ${\displaystyle b}$ and any real number exponent ${\displaystyle x}$. More involved definitions allow complex base and exponent, as well as certain types of matrices as base or exponent.

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 (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.

History

Antiquity

The Sand Reckoner

In The Sand Reckoner, Archimedes proved the law of exponents, 10^a · 10^b = 10^a+b, necessary to manipulate powers of 10. He then used powers of 10 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 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 (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 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 12² to represent 12x². 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 iii4 for 4x³.

"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.

Modern exponential notation

In 1636, James Hume used in essence modern notation, when in L'algèbre de Viète he wrote Aⁱⁱⁱ for A³. 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.

I designate ... aa, or a² in multiplying a by itself; and a³ in multiplying it once more again by a, and thus to infinity.

— René Descartes, 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 ax + bxx + cx³ + d.

"Indices"

Samuel Jeake introduced the term indices in 1696. The term involution was used synonymously with the term indices, but had declined in usage 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:

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.

20th century

As calculation was mechanized, notation was adapted to numerical capacity by conventions in exponential notation. For example Konrad Zuse introduced floating point arithmetic in his 1938 computer Z1. One register contained representation of leading digits, and a second contained representation of the exponent of 10. Earlier Leonardo Torres Quevedo contributed Essays on Automation (1914) which had suggested the floating-point representation of numbers. The more flexible decimal floating-point representation was introduced in 1946 with a Bell Laboratories computer. Eventually educators and engineers adopted scientific notation of numbers, consistent with common reference to order of magnitude in a ratio scale.

For instance, in 1961 the School Mathematics Study Group developed the notation in connection with units used in the metric system.

Exponents also came to be used to describe units of measurement and quantity dimensions. For instance, since force is mass times acceleration, it is measured in kg m/sec². Using M for mass, L for length, and T for time, the expression M L T^–2 is used in dimensional analysis to describe force.

Terminology

The expression b² = b · b is called "the square of b" or "b squared", because the area of a square with side-length b is b². (It is true that it could also be called "b to the second power", but "the square of b" and "b squared" are more traditional)

Similarly, the expression b³ = b · b · b is called "the cube of b" or "b cubed", because the volume of a cube with side-length b is b³.

When an exponent is a positive integer, that exponent indicates how many copies of the base are multiplied together. For example, 3⁵ = 3 · 3 · 3 · 3 · 3 = 243. The base 3 appears 5 times in the multiplication, because the exponent is 5. 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, so 3⁵ can be simply read "3 to the 5th", or "3 to the 5".

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

The base case is

${\displaystyle b^{1}=b}$

and the recurrence is

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

The associativity of multiplication implies that for any positive integers m and n,

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

and

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

Zero exponent

As mentioned earlier, a (nonzero) number raised to the 0 power is 1:

${\displaystyle b^{0}=1.}$

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

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

also holds for ${\displaystyle n=0}$.

The case of 0⁰ is controversial. In contexts where only integer powers are considered, the value 1 is generally assigned to 0⁰ 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 integer n and nonzero b:

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

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

This definition of exponentiation with negative exponents is the only one that allows extending the identity ${\displaystyle b^{m+n}=b^{m}\cdot b^{n}}$ to negative exponents (consider the case ${\displaystyle m=-n}$).

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 1 (for example, the square matrices of a given dimension). In particular, in such a structure, the inverse of an invertible element x is standardly denoted ${\displaystyle x^{-1}.}$

Identities and properties

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

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

Unlike addition and multiplication, exponentiation is not commutative: for example, ${\displaystyle 2^{3}=8}$, but reversing the operands gives the different value ${\displaystyle 3^{2}=9}$. Also unlike addition and multiplication, exponentiation is not associative: for example, (2³)² = 8² = 64, whereas 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,

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

which, in general, is different from

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

Powers of a sum

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

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

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

Combinatorial interpretation

For nonnegative integers n and m, the value of n^m is the number of functions from a set of m elements to a set of n elements (see cardinal exponentiation). Such functions can be represented as m-tuples from an n-element set (or as m-letter words from an n-letter alphabet). Some examples for particular values of m and n are given in the following table:

n^m	The n^m possible m-tuples of elements from the set {1, ..., n}
0^5 = 0	none
1^4 = 1	(1, 1, 1, 1)
2^3 = 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)
3^2 = 9	(1, 1), (1, 2), (1, 3), (2, 1), (2, 2), (2, 3), (3, 1), (3, 2), (3, 3)
4^1 = 4	(1), (2), (3), (4)
5^0 = 1	()

Particular bases

Powers of ten

In the base ten (decimal) number system, integer powers of 10 are written as the digit 1 followed or preceded by a number of zeroes determined by the sign and magnitude of the exponent. For example, 10³ = 1000 and 10⁻⁴ = 0.0001.

Exponentiation with base 10 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 as 2.99792458×10⁸ m/s and then approximated as 2.998×10⁸ m/s.

SI prefixes based on powers of 10 are also used to describe small or large quantities. For example, the prefix kilo means 10³ = 1000, so a kilometre is 1000 m.

Powers of two

The first negative powers of 2 have special names: ${\displaystyle 2^{-1}}$is a half; ${\displaystyle 2^{-2}}$ is a quarter.

Powers of 2 appear in set theory, since a set with n members has a power set, the set of all of its subsets, which has 2ⁿ members.

Integer powers of 2 are important in computer science. The positive integer powers 2ⁿ give the number of possible values for an n-bit integer binary number; for example, a byte may take 2⁸ = 256 different values. The binary number system expresses any number as a sum of powers of 2, and denotes it as a sequence of 0 and 1, separated by a binary point, where 1 indicates a power of 2 that appears in the sum; the exponent is determined by the place of this 1: the nonnegative exponents are the rank of the 1 on the left of the point (starting from 0), and the negative exponents are determined by the rank on the right of the point.

Powers of one

Every power of one equals: 1ⁿ = 1.

Powers of zero

For a positive exponent n > 0, the nth power of zero is zero: 0ⁿ = 0. For a negative\ exponent, ${\displaystyle 0^{-n}=1/0^{n}=1/0}$ is undefined.

The expression 0⁰ is either defined as ${\displaystyle \lim _{x\to 0}x^{x}=1}$, or it is left undefined.

Powers of negative one

Since a negative number times another negative is positive, we have:

${\displaystyle (-1)^{n}=\left\{{\begin{array}{rl}1&{\text{for even }}n,\\-1&{\text{for odd }}n.\\\end{array}}\right.}$

Because of this, powers of −1 are useful for expressing alternating sequences. For a similar discussion of powers of the complex number i, see § 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:

bⁿ → ∞ as n → ∞ when b > 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 as n → ∞ when |b| < 1

Any power of one is always one:

bⁿ = 1 for all n for b = 1

Powers of a negative number ${\displaystyle b\leq -1}$ alternate between positive and negative as n alternates between even and odd, and thus do not tend to any limit as n grows.

If the exponentiated number varies while tending to 1 as the exponent tends to infinity, then the limit is not necessarily one of those above. A particularly important case is

(1 + 1/n)ⁿ → e as n → ∞

See § 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 ${\displaystyle f(x)=cx^{n}}$, where ${\displaystyle c\neq 0}$, are sometimes called power functions. When ${\displaystyle n}$ is an integer and ${\displaystyle n\geq 1}$, two primary families exist: for ${\displaystyle n}$ even, and for ${\displaystyle n}$ odd. In general for ${\displaystyle c>0}$, when ${\displaystyle n}$ is even ${\displaystyle f(x)=cx^{n}}$ will tend towards positive infinity with increasing ${\displaystyle x}$, and also towards positive infinity with decreasing ${\displaystyle x}$. All graphs from the family of even power functions have the general shape of ${\displaystyle y=cx^{2}}$, flattening more in the middle as ${\displaystyle n}$ increases. Functions with this kind of symmetry (${\displaystyle f(-x)=f(x)}$) are called even functions.

When ${\displaystyle n}$ is odd, ${\displaystyle f(x)}$'s asymptotic behavior reverses from positive ${\displaystyle x}$ to negative ${\displaystyle x}$. For ${\displaystyle c>0}$, ${\displaystyle f(x)=cx^{n}}$ will also tend towards positive infinity with increasing ${\displaystyle x}$, but towards negative infinity with decreasing ${\displaystyle x}$. All graphs from the family of odd power functions have the general shape of ${\displaystyle y=cx^{3}}$, flattening more in the middle as ${\displaystyle n}$ increases and losing all flatness there in the straight line for ${\displaystyle n=1}$. Functions with this kind of symmetry (${\displaystyle f(-x)=-f(x)}$) are called odd functions.

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

Table of powers of decimal digits

n	n^2	n^3	n^4	n^5	n^6	n^7	n^8	n^9	n^10
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

Rational exponents

From top to bottom: x^1/8, x^1/4, x^1/2, x¹, x², x⁴, x⁸.

If x is a nonnegative real number, and n is a positive integer, ${\displaystyle x^{1/n}}$ or ${\displaystyle {\sqrt[{n}]{x}}}$ denotes the unique nonnegative real nth root of x, that is, the unique nonnegative real number y such that ${\displaystyle y^{n}=x.}$

If x is a positive real number, and ${\displaystyle {\frac {p}{q}}}$ is a rational number, with p and q > 0 integers, then ${\textstyle x^{p/q}}$ is defined as

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

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

If r is a positive rational number, 0^r = 0, by definition.

All these definitions are required for extending the identity ${\displaystyle (x^{r})^{s}=x^{rs}}$ 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 nth root, which is negative, if n is odd, and no real root if n is even. In the latter case, whichever complex nth root one chooses for ${\displaystyle x^{\frac {1}{n}},}$ the identity ${\displaystyle (x^{a})^{b}=x^{ab}}$ cannot be satisfied. For example,

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

See § Real exponents and § 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 (§ 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. One may choose one of these values, called the principal value, but there is no choice of the principal value for which the identity

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

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 of e^1/n is e⁰ = 1 when 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 b with an arbitrary real exponent x can be defined by continuity with the rule

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

where the limit is taken over rational values of r only. This limit exists for every positive b and every real x.

For example, if x = π, the non-terminating decimal representation 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 ${\displaystyle b^{\pi }:}$

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

So, the upper bounds and the lower bounds of the intervals form two sequences that have the same limit, denoted ${\displaystyle b^{\pi }.}$

This defines ${\displaystyle b^{x}}$ for every positive b and real x as a continuous function of b and x. See also Well-defined expression.

Exponential function

The exponential function may be defined as ${\displaystyle x\mapsto e^{x},}$ where ${\displaystyle e\approx 2.718}$ is Euler's number, but to avoid circular reasoning, this definition cannot be used here. Rather, we give an independent definition of the exponential function ${\displaystyle \exp(x),}$ and of ${\displaystyle e=\exp(1)}$, relying only on positive integer powers (repeated multiplication). Then we sketch the proof that this agrees with the previous definition: ${\displaystyle \exp(x)=e^{x}.}$

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

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

One has ${\displaystyle \exp(0)=1,}$ and the exponential identity (or multiplication rule) ${\displaystyle \exp(x)\exp(y)=\exp(x+y)}$ holds as well, since

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

and the second-order term ${\displaystyle {\frac {xy}{n^{2}}}}$ does not affect the limit, yielding ${\displaystyle \exp(x)\exp(y)=\exp(x+y)}$.

Euler's number can be defined as ${\displaystyle e=\exp(1)}$. It follows from the preceding equations that ${\displaystyle \exp(x)=e^{x}}$ when x is an integer (this results from the repeated-multiplication definition of the exponentiation). If x is real, ${\displaystyle \exp(x)=e^{x}}$ results from the definitions given in preceding sections, by using the exponential identity if x is rational, and the continuity of the exponential function otherwise.

The limit that defines the exponential function converges for every complex value of x, and therefore it can be used to extend the definition of ${\displaystyle \exp(z)}$, and thus ${\displaystyle e^{z},}$ from the real numbers to any complex argument 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 e^x as the exponential function allows defining b^x for every positive real numbers b, in terms of exponential and logarithm function. Specifically, the fact that the natural logarithm ln(x) is the inverse of the exponential function e^x means that one has

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

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

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

So, ${\displaystyle e^{x\ln b}}$ can be used as an alternative definition of b^x for any positive real 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 b is a positive real number, exponentiation with base b and complex exponent z is defined by means of the exponential function with complex argument (see the end of § Exponential function, above) as

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

where ${\displaystyle \ln b}$ denotes the natural logarithm of b.

This satisfies the identity

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

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

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

unless z is real or t is an integer.

Euler's formula,

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

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

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

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

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

Non-integer exponents with a complex base

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 nth roots, that is, of exponents ${\displaystyle 1/n,}$ where n is a positive integer. Although the general theory of exponentiation with non-integer exponents applies to nth 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 number z may be written in polar form as

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

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

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 nth root of a complex number can be obtained by taking the nth root of the absolute value and dividing its argument by n:

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

If ${\displaystyle 2\pi }$ is added to ${\displaystyle \theta }$, the complex number is not changed, but this adds ${\displaystyle 2i\pi /n}$ to the argument of the nth root, and provides a new nth root. This can be done n times, and provides the n nth roots of the complex number.

It is usual to choose one of the n nth root as the principal root. The common choice is to choose the nth root for which ${\displaystyle -\pi <\theta \leq \pi ,}$ that is, the nth root that has the largest real part, and, if there are two, the one with positive imaginary part. This makes the principal nth root a continuous function in the whole complex plane, except for negative real values of the radicand. This function equals the usual nth root for positive real radicands. For negative real radicands, and odd exponents, the principal nth root is not real, although the usual nth root is real. Analytic continuation shows that the principal nth root is the unique complex differentiable function that extends the usual nth 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 ${\displaystyle 2\pi ,}$ the complex number comes back to its initial position, and its nth roots are permuted circularly (they are multiplied by $e^{2i\pi /n}$). This shows that it is not possible to define a nth root function that is continuous in the whole complex plane.

Roots of unity

The three third roots of 1

The nth roots of unity are the n complex numbers such that wⁿ = 1, where n 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).

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

The nth roots of unity allow expressing all nth roots of a complex number z as the n products of a given nth roots of z with a nth root of unity.

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

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

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

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

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

for every z in its domain of definition.

Principal value

The principal value of the complex logarithm is the unique continuous function, commonly denoted ${\displaystyle \log ,}$ such that, for every nonzero complex number z,

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

and the argument of z satisfies

${\displaystyle -\pi <\operatorname {Arg} z\leq \pi .}$

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

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

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

If z is real and positive, the principal value of ${\displaystyle z^{w}}$ equals its usual value defined above. If ${\displaystyle w=1/n,}$ where n 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 ${\displaystyle \log z}$ and ${\displaystyle z^{w}}$ at the negative real values of z. In this case, it is useful to consider these functions as multivalued functions.

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

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

where k is any integer.

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

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

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

Computation

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

• Polar form of z. If ${\displaystyle z=a+ib}$ is the canonical form of z (a and b being real), then its polar form is $${\displaystyle z=\rho e^{i\theta }=\rho (\cos \theta +i\sin \theta ),}$$ with ${\textstyle \rho ={\sqrt {a^{2}+b^{2}}}}$ and ${\displaystyle \theta =\operatorname {atan2} (b,a)}$, where ⁠${\displaystyle \operatorname {atan2} }$⁠ is the two-argument arctangent function.
• Logarithm of z. The principal value of this logarithm is ${\displaystyle \log z=\ln \rho +i\theta ,}$ where ${\displaystyle \ln }$ denotes the natural logarithm. The other values of the logarithm are obtained by adding ${\displaystyle 2ik\pi }$ for any integer k.
• Canonical form of ${\displaystyle w\log z.}$ If ${\displaystyle w=c+di}$ with c and d real, the values of ${\displaystyle w\log z}$ are $${\displaystyle w\log z=(c\ln \rho -d\theta -2dk\pi )+i(d\ln \rho +c\theta +2ck\pi ),}$$ the principal value corresponding to ${\displaystyle k=0.}$
• Final result. Using the identities ${\displaystyle e^{x+y}=e^{x}e^{y}}$ and ${\displaystyle e^{y\ln x}=x^{y},}$ one gets $${\displaystyle z^{w}=\rho ^{c}e^{-d(\theta +2k\pi )}\left(\cos(d\ln \rho +c\theta +2ck\pi )+i\sin(d\ln \rho +c\theta +2ck\pi )\right),}$$ with ${\displaystyle k=0}$ for the principal value.

Examples

• ${\displaystyle i^{i}}$
  The polar form of i is ${\displaystyle i=e^{i\pi /2},}$ and the values of ${\displaystyle \log i}$ are thus $${\displaystyle \log i=i\left({\frac {\pi }{2}}+2k\pi \right).}$$ It follows that $${\displaystyle i^{i}=e^{i\log i}=e^{-{\frac {\pi }{2}}}e^{-2k\pi }.}$$So, all values of ${\displaystyle i^{i}}$ are real, the principal one being $${\displaystyle e^{-{\frac {\pi }{2}}}\approx 0.2079.}$$
• ${\displaystyle (-2)^{3+4i}}$
  Similarly, the polar form of −2 is ${\displaystyle -2=2e^{i\pi }.}$ So, the above described method gives the values $${\displaystyle {\begin{aligned}(-2)^{3+4i}&=2^{3}e^{-4(\pi +2k\pi )}(\cos(4\ln 2+3(\pi +2k\pi ))+i\sin(4\ln 2+3(\pi +2k\pi )))\\&=-2^{3}e^{-4(\pi +2k\pi )}(\cos(4\ln 2)+i\sin(4\ln 2)).\end{aligned}}}$$In this case, all the values have the same argument ${\displaystyle 4\ln 2,}$ and different absolute values.

In both examples, all values of ${\displaystyle z^{w}}$ have the same argument. More generally, this is true if and only if the real part of w 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 identity log(b^x) = x ⋅ log b holds whenever b is a positive real number and x is a real number. But for the principal branch of the complex logarithm one has

  $${\displaystyle \log((-i)^{2})=\log(-1)=i\pi \neq 2\log(-i)=2\log(e^{-i\pi /2})=2\,{\frac {-i\pi }{2}}=-i\pi }$$

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

  This identity does not hold even when considering log as a multivalued function. The possible values of log(w^z) contain those of z ⋅ log w as a proper subset. Using Log(w) for the principal value of log(w) and m, n as any integers the possible values of both sides are:

  $${\displaystyle {\begin{aligned}\left\{\log w^{z}\right\}&=\left\{z\cdot \operatorname {Log} w+z\cdot 2\pi in+2\pi im\mid m,n\in \mathbb {Z} \right\}\\\left\{z\log w\right\}&=\left\{z\operatorname {Log} w+z\cdot 2\pi in\mid n\in \mathbb {Z} \right\}\end{aligned}}}$$
• The identities (bc)^x = b^xc^x and (b/c)^x = b^x/c^x are valid when b and c are positive real numbers and x is a real number. But, for the principal values, one has $${\displaystyle (-1\cdot -1)^{\frac {1}{2}}=1\neq (-1)^{\frac {1}{2}}(-1)^{\frac {1}{2}}=i\cdot i=i^{2}=-1}$$ and $${\displaystyle \left({\frac {1}{-1}}\right)^{\frac {1}{2}}=(-1)^{\frac {1}{2}}=i\neq {\frac {1^{\frac {1}{2}}}{(-1)^{\frac {1}{2}}}}={\frac {1}{i}}=-i}$$ On the other hand, when x 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 incorrect.
• The identity (e^x)^y = e^xy holds for real numbers x and y, but assuming its truth for complex numbers leads to the following paradox, discovered in 1827 by Clausen: For any integer n, we have:
  1. ${\displaystyle e^{1+2\pi in}=e^{1}e^{2\pi in}=e\cdot 1=e}$
  2. ${\displaystyle \left(e^{1+2\pi in}\right)^{1+2\pi in}=e\qquad }$ (taking the ${\displaystyle (1+2\pi in)}$-th power of both sides)
  3. ${\displaystyle e^{1+4\pi in-4\pi ^{2}n^{2}}=e\qquad }$ (using ${\displaystyle \left(e^{x}\right)^{y}=e^{xy}}$ and expanding the exponent)
  4. ${\displaystyle e^{1}e^{4\pi in}e^{-4\pi ^{2}n^{2}}=e\qquad }$ (using ${\displaystyle e^{x+y}=e^{x}e^{y}}$)
  5. ${\displaystyle e^{-4\pi ^{2}n^{2}}=1\qquad }$ (dividing by e)
  but this is false when the integer n is nonzero. The error is the following: by definition, ${\displaystyle e^{y}}$ is a notation for ${\displaystyle \exp(y),}$ a true function, and ${\displaystyle x^{y}}$ is a notation for ${\displaystyle \exp(y\log x),}$ which is a multi-valued function. Thus the notation is ambiguous when x = e. Here, before expanding the exponent, the second line should be $${\displaystyle \exp \left((1+2\pi in)\log \exp(1+2\pi in)\right)=\exp(1+2\pi in).}$$ Therefore, when expanding the exponent, one has implicitly supposed that ${\displaystyle \log \exp z=z}$ for complex values of z, which is wrong, as the complex logarithm is multivalued. In other words, the wrong identity (e^x)^y = e^xy must be replaced by the identity $${\displaystyle \left(e^{x}\right)^{y}=e^{y\log e^{x}},}$$ which is a true identity between multivalued functions.

Irrationality and transcendence

If b is a positive real algebraic number, and x is a rational number, then b^x is an algebraic number. This results from the theory of algebraic extensions. This remains true if b is any algebraic number, in which case, all values of b^x (as a multivalued function) are algebraic. If x is irrational (that is, not rational), and both b and x are algebraic, Gelfond–Schneider theorem asserts that all values of b^x are transcendental (that is, not algebraic), except if b equals 0 or 1.

In other words, if x is irrational and ${\displaystyle b\not \in \{0,1\},}$ then at least one of b, x and b^x is transcendental.

Integer powers in algebra

The definition of exponentiation with positive integer exponents as repeated multiplication may apply to any associative operation denoted as a multiplication. The definition of x⁰ 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 element x is defined inductively by

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

If n is a negative integer, ${\displaystyle x^{n}}$ is defined only if x has a multiplicative inverse. In this case, the inverse of x is denoted x⁻¹, and xⁿ is defined as ${\displaystyle \left(x^{-1}\right)^{-n}.}$

Exponentiation with integer exponents obeys the following laws, for x and y in the algebraic structure, and m and n integers:

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

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, if f is a real function whose valued can be multiplied, ${\displaystyle f^{n}}$ denotes the exponentiation with respect of multiplication, and ${\displaystyle f^{\circ n}}$ may denote exponentiation with respect of function composition. That is,

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

and

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

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

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, if G is a group, ${\displaystyle x^{n}}$ is defined for every ${\displaystyle x\in G}$ and every integer n.

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 element x is the cyclic group generated by x. If all the powers of x are distinct, the group is isomorphic to the additive group ${\displaystyle \mathbb {Z} }$ of the integers. Otherwise, the cyclic group is finite (it has a finite number of elements), and its number of elements is the order of x. If the order of x is n, then ${\displaystyle x^{n}=x^{0}=1,}$ and the cyclic group generated by x consists of the n first powers of x (starting indifferently from the exponent 0 or 1).

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, g^h = 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 ${\displaystyle (g^{h})^{k}=g^{hk}}$ and ${\displaystyle (gh)^{k}=g^{k}h^{k}.}$

In a ring

In a ring, it may occur that some nonzero elements satisfy ${\displaystyle x^{n}=0}$ for some integer n. Such an element is said to be 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 ${\displaystyle x\neq 0}$ implies ${\displaystyle x^{n}\neq 0}$ for every positive integer n), the commutative ring is said to be reduced. Reduced rings are important in algebraic geometry, since the coordinate ring of an affine algebraic set is always a reduced ring.

More generally, given an ideal I in a commutative ring R, the set of the elements of R that have a power in I is an ideal, called the radical of I. The nilradical is the radical of the zero ideal. A radical ideal is an ideal that equals its own radical. In a polynomial ring ${\displaystyle k[x_{1},\ldots ,x_{n}]}$ over a field k, 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 ${\displaystyle A^{0}}$ is defined to be the identity matrix, and if A is invertible, then ${\displaystyle A^{-n}=\left(A^{-1}\right)^{n}}$.

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 ${\displaystyle A^{2}x}$ is the state of the system after two time steps, and so forth: ${\displaystyle A^{n}x}$ is the state of the system after n time steps. The matrix power ${\displaystyle A^{n}}$ 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, ${\displaystyle d/dx}$, which is a linear operator acting on functions ${\displaystyle f(x)}$ to give a new function ${\displaystyle (d/dx)f(x)=f'(x)}$. The nth power of the differentiation operator is the nth derivative:

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

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

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 of 0. Common examples are the field of complex numbers, the real numbers and the rational numbers, considered earlier in this article, which 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 ${\displaystyle q=p^{k},}$ where p is a prime number, and k is a positive integer. For every such q, there are fields with q elements. The fields with q elements are all isomorphic, which allows, in general, working as if there were only one field with q elements, denoted ${\displaystyle \mathbb {F} _{q}.}$

One has

${\displaystyle x^{q}=x}$

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

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

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

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

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