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- Exponentiation

**Exponentiation** is a mathematical operation, written as, involving two numbers, the *base* and the *exponent* or *power*, and pronounced as " raised to the power of ".^{[1]} ^{[2]} When is a positive integer, exponentiation corresponds to repeated multiplication of the base: that is, is the product of multiplying bases:

*b*^{n}=*\underbrace{b* x ... x *b}*_{n}*.*

The exponent is usually shown as a superscript to the right of the base. In that case, is called "*b* raised to the *n*th power", "*b* raised to the power of *n*", "the *n*th power of *b*", "*b* to the *n*th power",^{[3]} or most briefly as "*b* to the *n*th".

One has, and, for any positive integers and, one has . To extend this property to non-positive integer exponents, is defined to be, and (with a positive integer and not zero) is defined as . In particular, is equal to, the *reciprocal* of .

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.

The term *power* (Latin: *potentia, potestas, dignitas*) is a mistranslation^{[4]} ^{[5]} of the ancient Greek δύναμις (*dúnamis*, here: "amplification"^{[4]}) used by the Greek mathematician Euclid for the square of a line, following Hippocrates of Chios.^{[6]} In *The Sand Reckoner*, Archimedes discovered and proved the law of exponents,, necessary to manipulate powers of . 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.^{[7]}

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.^{[8]} ^{[9]} 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).^{[10]} *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.^{[11]}

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, as .

Another historical synonym, **involution**, is now rare^{[12]} 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 expression is called "the square of *b*" or "*b* squared", because the area of a square with side-length is .

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

When it is a positive integer, the exponent indicates how many copies of the base are multiplied together. For example, . The base appears times in the multiplication, because the exponent is . Here, 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 can be simply read "3 to the 5th", or "3 to the 5". Therefore, the exponentiation can be expressed as "*b* to the power of *n*", "*b* to the *n*th power", "*b* to the *n*th", or most briefly as "*b* to the *n*".

A formula with nested exponentiation, such as (which means and not), is called a **tower of powers**, or simply a **tower**.

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

The definition of the exponentiation as an iterated multiplication can be formalized by using induction,^{[13]} and this definition can be used as soon one has an associative multiplication:

The base case is

*b*^{1}=*b*

*b*^{n+1}=*b*^{n} ⋅ *b.*

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

*b*^{m+n}=*b*^{m} ⋅ *b*^{n,}

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

By definition, any nonzero number raised to the power is :^{[14]}

*b*^{0=1.}

This definition is the only possible that allows extending the formula

*b*^{m+n}=*b*^{m ⋅ }*b*^{n}

Intuitionally,

*b*^{0}

*b*^{0=1}

The case of is more complicated. In contexts where only integer powers are considered, the value is generally assigned to

0^{0,}

Exponentiation with negative exponents is defined by the following identity, which holds for any integer and nonzero :

*b*^{-n}=

1 | |

b^{n} |

*.*

inf*ty*

This definition of exponentiation with negative exponents is the only one that allows extending the identity

*b*^{m+n}=*b*^{m ⋅ }*b*^{n}

*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 (for example, the square matrices of a given dimension). In particular, in such a structure, the inverse of an invertible element is standardly denoted

*x*^{-1}*.*

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

*\begin{align}
**b*^{m}*&*=*b*^{m} ⋅ *b*^{n}*\\
**\left(b*^{m\right)}^{n}*&*=*b*^{m}*\\
**(b* ⋅ *c)*^{n}*&*=*b*^{n} ⋅ *c*^{n
\end{align}}

Unlike addition and multiplication, exponentiation is not commutative. For example, . Also unlike addition and multiplication, exponentiation is not associative. For example,, whereas . 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,

p^{q} | |

b |

=

\left(p^{q\right)} | |

b |

*,*

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

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

n | |

(a+b) | |

i=0 |

*\binom{n}{i}a*^{ib}^{n-i}

n | |

=\sum | |

i=0 |

n! | |

i!(n-i)! |

*a*^{ib}^{n-i}*.*

However, this formula is true only if the summands commute (i.e. that), which is implied if they belong to a structure that is commutative. Otherwise, if and 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**.

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

The possible -tuples of elements from the set | ||
---|---|---|

0 = 0 | ||

1 = 1 | (1, 1, 1, 1) | |

2 = 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 = 9 | (1, 1), (1, 2), (1, 3), (2, 1), (2, 2), (2, 3), (3, 1), (3, 2), (3, 3) | |

4 = 4 | (1), (2), (3), (4) | |

5 = 1 |

See also: Scientific notation.

See main article: Power of 10. In the base ten (decimal) number system, integer powers of are written as the digit followed or preceded by a number of zeroes determined by the sign and magnitude of the exponent. For example, and .

Exponentiation with base is used in scientific notation to denote large or small numbers. For instance, (the speed of light in vacuum, in metres per second) can be written as and then approximated as .

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

See main article: Power of two. The first negative powers of are commonly used, and have special names, e.g.: *half* and *quarter*.

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

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

The powers of one are all one: .

If the exponent is positive, the th power of zero is zero: .

If the exponent is negative, the th power of zero is undefined, because it must equal

1*/*0^{-n}

1*/*0

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

If is an even integer, then .

If is an odd integer, then .

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

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

as when

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:

as when

Any power of one is always one:

for all if

Powers of alternate between and as alternates between even and odd, and thus do not tend to any limit as grows.

If,, alternates between larger and larger positive and negative numbers as alternates between even and odd, and thus does not tend to any limit as grows.

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

as

See below.

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

Real functions of the form

*f(x)*=*cx*^{n}

*c**\ne*0

*n*

*n**\ge*1

*n*

*n*

*c**>*0

*n*

*f(x)*=*cx*^{n}

*x*

*x*

*y*=*cx*^{2}

*n*

When

*n*

*f(x)*

*x*

*x*

*c**>*0

*f(x)*=*cx*^{n}

*x*

*x*

*y*=*cx*^{3}

*n*

*n*=1

For

*c**<*0

n | n^{2} | n^{3} | n^{4} | n^{5} | n^{6} | n^{7} | n^{8} | n^{9} | n^{10} | |
---|---|---|---|---|---|---|---|---|---|---|

2 | 4 | 8 | 16 | 32 | 64 | 128 | 256 | 512 | 1024 | |

3 | 9 | 27 | 81 | 243 | 729 | |||||

4 | 16 | 64 | 256 | 1024 | ||||||

5 | 25 | 125 | 625 | 3125 | ||||||

6 | 36 | 216 | 1296 | |||||||

7 | 49 | 343 | 2401 | |||||||

8 | 64 | 512 | 4096 | |||||||

9 | 81 | 729 | 6561 | |||||||

10 | 100 | 1000 |

If is a nonnegative real number, and is a positive integer,

| ||||

x |

*\sqrt[n]x*

*y*^{n=x.}

If is a positive real number, and

pq | |

| ||||

x |

^{p\right)}

| |||||||||||

| ||||

y=x |

| ||||

(x |

^{p=\left((y}^{p)}^{q\right)}

| ||||

^{p\right)}

| ||||

| ||||

If is a positive rational number,

0^{r=0,}

All these definitions are required for extending the identity

*(x*^{r)}^{s}=*x*^{rs}

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 th root, which is negative if is odd, and no real root if is even. In the latter case, whichever complex th root one chooses for

| ||||

x |

*(x*^{a)}^{b=x}^{ab}

*\left((*-1*)*^{2\right)}

| ||||

| ||||

1 |

| ||||

(-1) |

=*(*-1*)*^{1=-1.}

See and 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 (below), or in terms of the logarithm of the base and the exponential function (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). 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*^{r}

Since any irrational number can be expressed as the limit of a sequence of rational numbers, exponentiation of a positive real number with an arbitrary real exponent can be defined by continuity with the rule^{[16]}

*b*^{x}=*\lim*_{r}*b*^{r} *(b**\in*R^{+,}*x**\in*R*),*

For example, if, the non-terminating decimal representation 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*

*b*^{\pi.}

This defines

*b*^{x}

See main article: Exponential function. The *exponential function* is often defined as

*x\mapsto**e*^{x,}

*e* ≈ 2*.*718

*\exp(x),*

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

There are many equivalent ways to define the exponential function, one of them being to define it as the inverse function of the natural logarithm. Precisely, the natural logarithm is the antiderivative of

1*/x*

ln

x | |

x=\int | |

1 |

dt | |

t. |

*\exp(*0*)*=1*,*

*\exp(x*+*y)*=*\exp(x)\exp(y)*

Euler's number can be defined as

*e*=*\exp(*1*)*

*\exp(x)*=*e*^{x}

*\exp(x)*=*e*^{x}

The exponential function satisfies the equation

infty | |

\exp(x)=\sum | |

n=0 |

x^{n} | |

n! |

*.*

*e*^{z,}

The definition of as the exponential function allows defining for every positive real numbers, in terms of exponential and logarithm function. Specifically, the fact that the natural logarithm is the inverse of the exponential function means that one has

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

*(e*^{x)}^{y=e}^{xy}*,*

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

So,

*e*^{x}

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

*b*^{z}=*e*^{(zln}*,*

ln*b*

This satisfies the identity

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

*\left(b*^{z\right)}^{t}*\ne**b*^{zt}*,*

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

*b*^{z}

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

*b*^{x+iy}=*b*^{x}*b*^{iy}=*b*^{x}*e*^{iyln}=*b*^{x(\cos(yln}*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 of th roots, that is, of exponents

1*/n,*

Every nonzero complex number may be witten in polar form as

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

*\rho*

*\theta*

*\theta*

*\theta*+2*k\pi*

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

*\left(\rho**e*^{i\theta}

| ||||

\right) |

| ||||

If

2*i\pi*

*\theta,*

2*i\pi/n*

It is usual to choose one of the th root as the principal root. The common choice is to choose the th root for which

-*\pi<\theta\le**\pi,*

If the complex number is moved around zero by increasing its argument, after an increment of

2*\pi,*

See main article: Root of unity.

The th roots of unity are the complex numbers such that, where 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 th roots of unity are the first powers of

*\omega*

| ||||

=e |

1=*\omega*^{0=\omega}^{n,}*\omega*=*\omega*^{1,}*\omega*^{2,}*\omega*^{n-1}*.*

*\omega*^{k=e}

| ||||

*,*

-1*;*

*i*

-*i.*

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

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

As the number

| ||||

e |

1^{1/n}

Defining exponentiation with complex bases leads to difficulties that are similar to those described in the preceding section, except that there are, in general, there are infinitely many possible values for z^w. So, either a principal value is defined, which is not continuous for the values of 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}^{wlog}*,*

log*z*

*e*^{log}=*z*

The principal value of the complex logarithm is the unique function, commonly denoted

log*,*

*e*^{log}=*z,*

-*\pi**<Im**\le**\pi.*

*z*=0*,*

log*z*=ln*z.*

The principal value of

*z*^{w}

*z*^{w=e}^{wlog}*,*

log*z*

The function

*(z,w)\to**z*^{w}

If is real and positive, the principal value of

*z*^{w}

*w*=1*/n,*

In some contexts, there is a problem with the discontinuity of the principal values of

log*z*

*z*^{w}

If

log*z*

2*ik\pi*+log*z,*

*z*^{w}

*e*^{w(2ik\pi}=*z*^{we}^{2ik\pi}*,*

Different values of give different values of

*z*^{w}

*e*^{a=e}^{b}

*a*-*b*

2*\pi.*

If

w= | mn |

*n>*0*,*

*z*^{w}

*m*=1*,*

The multivalued exponentiation is holomorphic for

*z\ne*0*,*

*z*^{w}

The *canonical form*

*x*+*iy*

*z*^{w}

*Polar form of*. If

*z*=*a*+*ib*

*\rho*=*\sqrt{a*^{2+b}^{2}}

*\theta*=*\operatorname{atan*2*}(a,b)*

*Logarithm of*. The principal value of this logarithm is

log*z*=ln*\rho*+*i\theta,*

ln

2*ik\pi*

*Canonical form of*

*w*log*z.*

*w*=*c*+*di*

*w*log*z*

*k*=0*.*

*Final result.*Using the identities

*e*^{x+y}*e*^{x}=*e*^{y}

*e*^{yln}=*x*^{y,}

*k*=0

*i*^{i}

The polar form of is

*i*=*e*^{i\pi/2}*,*

log*i*

*i*^{i}

*(*-2*)*^{3+4i}

Similarly, the polar form of is

-2=2*e*^{i}*.*

4ln2*,*

In both examples, all values of

*z*^{w}

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:

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

In other words, if is irrational and

*b\not\in**\{*0*,*1*\},*

The definition of exponentiation with positive integer exponents as repeated multiplication may apply to any associative operation denoted as a multiplication.^{[20]} The definition of

*x*^{0}

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 is defined inductively by

*x*^{0}=1*,*

*x*^{n+1}=*x**x*^{n}

If is a negative integer,

*x*^{n}

*x*^{-1}*,*

*x*^{n}

*\left(x*^{-1}*\right)*^{-n}*.*

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

*\begin{align}
x*^{0&=1\\
x}^{m+n}*&*=*x*^{m}*x*^{n\\
(x}^{m)}^{n&=x}^{mn}*\\
(xy)*^{n&=x}^{n}*y*^{n} if*xy*=*yx,*and,inparticular,ifthemultiplicationiscommutative.*
\end{align}*

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 is a real function whose valued can be multiplied,

*f*^{n}

*f*^{\circ}

*(f*^{n)(x)=(f(x))}^{n=f(x)}*f(x)* … *f(x),*

*(f*^{\circ}*)(x)*=*f(f(* … *f(f(x))* … *)).*

*(f*^{n)(x)}

*f(x)*^{n,}

*(f*^{\circ}*)(x)*

*f*^{n(x).}

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 is a group,

*x*^{n}

*x\in**G*

*\Z*

*x*^{n=x}^{0=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,, 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}

*(gh)*^{k=g}^{kh}^{k.}

In a ring, it may occurs that some nonzero elements satisfy

*x*^{n=0}

If the nilradical is reduced to the zero ideal (that is, if

*x* ≠ 0

*x*^{n ≠ }0

*k[x*_{1,}*\ldots,**x*_{n]}

If *A* is a square matrix, then the product of *A* with itself *n* times is called the matrix power. Also

*A*^{0}

*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.^{[24]} This is the standard interpretation of a Markov chain, for example. Then

*A*^{2x}

*A*^{nx}

*A*^{n}

Apart from matrices, more general linear operators can also be exponentiated. An example is the derivative operator of calculus,

*d/dx*

*f(x)*

*(d/dx)f(x)*=*f'(x)*

\left( | d |

dx |

*\right)*^{nf(x)}=

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.^{[25]} 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.

See 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 of . 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,}

F_{q.}

One has

*x*^{q=x}

*x\in*F_{q.}

A primitive element in

F_{q}

*\{g*^{1=g,}*g*^{2,}*\ldots,**g*^{p-1}=*g*^{0=1\}}

F_{q.}

*\varphi**(p*-1*)*

F_{q,}

*\varphi*

In

F_{q,}

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

*x*^{p=x}

F_{q,}

*\begin{align}
F\colon{}**&*F_{q}*\to*F_{q\\
&}*x\mapsto**x*^{p
\end{align}}

F_{q,}

*q*=*p*^{k,}

F_{q}

F_{q}

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, if is a primitive element in

F_{q,}

*g*^{e}

*g*^{e}

The Cartesian product of two sets and is the set of the ordered pairs

*(x,y)*

*x\in**S*

*y\in**T.*

*(x,(y,z)),*

*((x,y),z),*

*(x,y,z).*

This allows defining the th power

*S*^{n}

*(x*_{1,}*\ldots,**x*_{n)}

When is endowed with some structure, it is frequent that

*S*^{n}

*\R*^{n}

*\R*

*\R,*

A -tuple

*(x*_{1,}*\ldots,**x*_{n)}

*\{*1*,\ldots,**n\}.*

Given two sets and, the set of all functions from to is denoted

*S*^{T}

*(S*^{T)}^{U\cong}*S*^{T x }*,*

*S*^{T\sqcup}*\cong**S*^{T x }*S*^{U,}

x

*\sqcup*

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,

*\R*^{\N}

*\R*^{(\N)}

In this context, can represents the set

*\{*0*,*1*\}.*

2^{S}

*\{*0*,*1*\},*

This fits in with the exponentiation of cardinal numbers, in the sense that, where is the cardinality of .

See main article: Cartesian closed category. In the category of sets, the morphisms between sets and are the functions from to . It results that the set of the functions from to that is denoted

*Y*^{X}

*\hom(X,Y).*

*(S*^{T)}^{U\cong}*S*^{T x }

*\hom(U,S*^{T)\cong}*\hom(T* x *U,S).*

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}

*Y\to**T* x *Y.*

*Y\to**X* x *Y*

See main article: 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, the functions addition, multiplication, exponentiation, and tetration yield 6, 9, 27, and respectively.

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

More precisely, consider the function defined on Then can be viewed as a subset of (that is, the set of all pairs with, belonging to the extended real number line, endowed with the product topology), which will contain the points at which the function has a limit.

In fact, has a limit at all accumulation points of, except for,, and .^{[26]} Accordingly, this allows one to define the powers by continuity whenever,, except for 0^{0}, (+∞)^{0}, 1^{+∞} and 1^{−∞}, which remain indeterminate forms.

Under this definition by continuity, we obtain:

- and, when .
- and, when .
- and, when .
- and, when .

These powers are obtained by taking limits of for *positive* values of . This method does not permit a definition of when, since pairs with are not accumulation points of .

On the other hand, when is an integer, the power is already meaningful for all values of, including negative ones. This may make the definition obtained above for negative problematic when is odd, since in this case as tends to through positive values, but not negative ones.

Computing *b*^{n} using iterated multiplication requires multiplication operations, but it can be computed more efficiently than that, as illustrated by the following example. To compute 2^{100}, 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))}

2^{2} = 4 | |

2 (2^{2}) = 2^{3} = 8 | |

(2^{3})^{2} = 2^{6} = 64 | |

(2^{6})^{2} = 2^{12} = | |

(2^{12})^{2} = 2^{24} = | |

2 (2^{24}) = 2^{25} = | |

(2^{25}) ^{2} 2 = 2^{50} = | |

(2^{25})^{2} = 2^{100} = |

In general, the number of multiplication operations required to compute can be reduced to

*\sharp**n*+*\lfloor*log_{2}*n\rfloor*-1*,*

*\sharp**n*

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,*

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

If the domain of a function equals its codomain, one may compose the function with itself an arbitrary number of time, and this defines the th power of the function under composition, commonly called the *th iterate* of the function. Thus

*f*^{n}

*f*^{3(x)}

*f(f(f(x))).*

When a multiplication is defined on the codomain of the function, this defines a multiplication on functions, the pointwise multiplication, which induces another exponentiation. When using functional notation, the two kinds of exponentiation are generaly distinguished by placing the exponent of the functional iteration *before* the parentheses enclosing the arguments of the function, and placing the exponent of pointwise multiplication *after* the parentheses. Thus

*f*^{2(x)=}*f(f(x)),*

*f(x)*^{2=}*f(x)* ⋅ *f(x).*

*f*^{\circ}=*f\circ**f**\circ**f,*

*f*^{3=f ⋅ }*f* ⋅ *f.*

*\sin*^{2}*x*

*\sin*^{2(x)}

*\sin(x)* ⋅ *\sin(x)*

*\sin(\sin(x)),*

In this context, the exponent

-1

*\sin*^{-1}*x*=*\sin*^{-1}*(x)*=*\arcsin**x.*

1/\sin(x)= | 1{\sin |

x}. |

Programming languages generally express exponentiation either as an infix operator or as a (prefix) function, as they are linear notations which do not support superscripts:

`x ↑ y`

: Algol, Commodore BASIC, TRS-80 Level II/III BASIC.^{[28]}^{[29]}

- Web site: 2020-03-01. Compendium of Mathematical Symbols. 2020-08-27. Math Vault. en-US.
- Web site: Nykamp. Duane. Basic rules for exponentiation. August 27, 2020. Math Insight.
- Web site: Weisstein. Eric W.. Power. 2020-08-27. mathworld.wolfram.com. en.
- Book: Rotman, Joseph J.. Joseph J. Rotman. 2015. Advanced Modern Algebra, Part 1. Providence, RI. American Mathematical Society. p. 130, fn. 4. 978-1-4704-1554-9. 3rd. Graduate Studies in Mathematics. 165.
- Book: Szabó, Árpád. 1978. The Beginnings of Greek Mathematics. Dordrecht. D. Reidel. 37. 90-277-0819-3. Synthese Historical Library. 17. A.M. Ungar.
- Book: Ball, W. W. Rouse. W. W. Rouse Ball. 1915. A Short Account of the History of Mathematics. London. Macmillan. 38. 6th.
- Book: Cajori, Florian. Florian Cajori. 1928. A History of Mathematical Notations. London. Open Court Publishing Company. 344. 1.
- http://jeff560.tripod.com/e.html Earliest Known Uses of Some of the Words of Mathematics
- Book: Stifel, Michael. Michael Stifel. 1544. Arithmetica integra. Nuremberg. Johannes Petreius. 235v.
- Web site: Zenzizenzizenzic. World Wide Words. Michael. Quinion. Michael Quinion. 2020-04-16.
- Book: Descartes, René. René Descartes. 1637. Discourse de la méthode [...]]. Leiden. Jan Maire. 299.
*La Géométrie*.*Et*aa*, ou*aa^{2}, pour multiplier*par soy mesme; Et*aa^{3}, pour le multiplier encore une fois par*, & ainsi a l'infini*. (And*aa*, or*a*^{2}, in order to multiply*a*by itself; and*a*^{3}, in order to multiply it once more by*a*, and thus to infinity). - The most recent usage in this sense cited by the OED is from 1806 .
- Book: Abstract Algebra: an inquiry based approach . Jonathan K. . Hodge . Steven . Schlicker . Ted . Sundstorm . 94 . 2014 . CRC Press . 978-1-4665-6706-1.
- Book: Technical Shop Mathematics . Thomas . Achatz . 101 . 2005 . 3rd . Industrial Press . 978-0-8311-3086-2.
- Book: Anton. Howard. Bivens. Irl. Davis. Stephen. Calculus: Early Transcendentals. 2012. limited. John Wiley & Sons. 28. 9780470647691. 9th.
- Book: Denlinger, Charles G. . Elements of Real Analysis . Jones and Bartlett . 2011 . 278–283 . 978-0-7637-7947-4.
- Book: Introduction to Algorithms . second . Thomas H. . Cormen . Charles E. . Leiserson . Ronald L. . Rivest . Clifford . Stein . . 2001 . 978-0-262-03293-3. Online resource
- Book: Difference Equations: From Rabbits to Chaos . Difference Equations: From Rabbits to Chaos . . Paul . Cull . Mary . Flahive . Mary Flahive . Robby . Robson . 2005 . Springer . 978-0-387-23234-8. Defined on p. 351
- "Principal root of unity", MathWorld.
- More generally, power associativity is sufficient for the definition.
- Book: Nicolas . Bourbaki. Algèbre. 1970. Springer., I.2
- Book: David M. . Bloom . Linear Algebra and Geometry . registration . 1979 . 978-0-521-29324-2 . 45.
- Chapter 1, Elementary Linear Algebra, 8E, Howard Anton
- , Chapter 5.
- E. Hille, R. S. Phillips:
*Functional Analysis and Semi-Groups*. American Mathematical Society, 1975. - Nicolas Bourbaki,
*Topologie générale*, V.4.2. - Gordon . D. M. . 10.1006/jagm.1997.0913 . A Survey of Fast Exponentiation Methods . Journal of Algorithms . 27 . 129–146 . 1998 . 10.1.1.17.7076 .
- News: BASCOM - A BASIC compiler for TRS-80 I and II . Timothy "Tim" A. . Daneliuk . 1982-08-09 . . Software Reviews . . 4 . 31 . 41–42 . 2020-02-06 . live . https://web.archive.org/web/20200207104336/https://books.google.de/books?id=NDAEAAAAMBAJ&pg=PA42&lpg=PA42&focus=viewport&dq=TRS-80+exponention&hl=de#v=onepage&q=TRS-80%20exponention&f=false . 2020-02-07 . [...] If [...] squaring is accomplished with TRS-80 BASIC's exponentiation (up-arrow) function, interpreter run time is 22 minutes 20 seconds, and compiled run time is 20 minutes 3 seconds. [...] -->.
- 80 Contents . . . 0744-7868 . October 1983 . 45 . 5 . 2020-02-06 . [...] The left bracket, [, replaces the up arrow used by [[RadioShack]] to indicate exponentiation on our printouts. When entering programs published in 80 Micro, you should make this change. [...]. (NB. At code point 5Bh the TRS-80 character set has an up-arrow symbol "↑" in place of the ASCII left square bracket "[".) -->}}</ref>
* <code>x ^ y</code>: [[AWK]], BASIC, J, MATLAB, Wolfram Language (Mathematica), R, Microsoft Excel, Analytica, TeX (and its derivatives), TI-BASIC, bc (for integer exponents), Haskell (for nonnegative integer exponents), Lua and most computer algebra systems. Conflicting uses of the symbol
`^`

include: XOR (in POSIX Shell arithmetic expansion, AWK, C, C++, C#, D, Go, Java, JavaScript, Perl, PHP, Python, Ruby and Tcl), indirection (Pascal), and string concatenation (OCaml and Standard ML).`x ^^ y`

: Haskell (for fractional base, integer exponents), D.`x ** y`

: Ada, Z shell, KornShell, Bash, COBOL, CoffeeScript, Fortran, FoxPro, Gnuplot, Groovy, JavaScript, OCaml, F#, Perl, PHP, PL/I, Python, Rexx, Ruby, SAS, Seed7, Tcl, ABAP, Mercury, Haskell (for floating-point exponents), Turing, VHDL.`pown x y`

: F# (for integer base, integer exponent).`x⋆y`

: APL.

Many other programming languages lack syntactic support for exponentiation, but provide library functions:

`pow(x, y)`

: C, C++.`Math.Pow(x, y)`

: C#.`math:pow(X, Y)`

: Erlang.`Math.pow(x, y)`

: Java.`[Math]::Pow(x, y)`

: PowerShell.`(expt x y)`

: Common Lisp.

For certain exponents there are special ways to compute

*x*^{y}much faster than through generic exponentiation. These cases include small positive and negative integers (prefer*x*·*x*over*x*^{2}; prefer 1/*x*over*x*^{−1}) and roots (prefer sqrt(*x*) over*x*^{0.5}, prefer cbrt(*x*) over*x*^{1/3}).Not all programming languages adhere to the same association convention for exponentiation: while the Wolfram language, Google Search and others use right-association (i.e.

`a^b^c`

is evaluated as`a^(b^c)`

), many computer programs such as Microsoft Office Excel and Matlab associate to the left (i.e.`a^b^c`

is evaluated as`(a^b)^c`

).## See also

- Double exponential function
- Exponential decay
- Exponential field
- Exponential growth
- List of exponential topics
- Modular exponentiation
- Scientific notation
- Unicode subscripts and superscripts
*x*^{y}=*y*^{x}- Zero to the power of zero

## References