# Dictionary Definition

exponentiation n : the process of raising a
quantity to some assigned power [syn: involution]

# User Contributed Dictionary

## English

### Noun

- In the context of "mathematics|arithmetic|uncountable": The
process of calculating a power by multiplying together a
number of equal factors, where the exponent specifies the number
of factors to multiply.
- In 54, the factors are all 5, the power is 4 and exponentiation gives the result 625.

- In the context of "mathematics|arithmetic|countable": A mathematical problem involving exponentiation.

#### Translations

process

- Czech: mocnění
- French: exponentiation
- Swedish: exponentiering

problem

## French

### Noun

exponentiation- exponentiation (process)

# Extensive Definition

Exponentiation is a mathematical operation,
written an, involving two numbers, the base a
and the exponent n. When n'' is a positive
integer, exponentiation
corresponds to repeated multiplication:

- a^n = \underbrace_n,

just as multiplication by a whole number
corresponds to repeated addition:

- a \times n = \underbrace_n.

The exponent is usually shown as a superscript to the right of
the base. The exponentiation an can be read as: a raised to the
n-th power or a raised to the power [of] n, or more briefly: a to
the n-th power or a to the power [of] n, or even more briefly: a to
the n. Some exponents can be read in a certain way; for example a2
is usually read as a squared and a3 as a cubed.

The power an can also be defined when the
exponent n is a negative integer. When the base a is a positive
real number, exponentiation is defined for real and even complex
exponents n. The special exponential
function ex is fundamental for this definition. It enables the
functions of trigonometry to be
expressed by exponentiation. However, when the base a is not a
positive real number and the exponent n is not an integer, then an
cannot be defined as a unique continuous
function of a.

Exponentiation where the exponent is a matrix
is used for solving systems of
linear differential equations.

Exponentiation is used pervasively in many other
fields as well, 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.

## Exponentiation with integer exponents

The exponentiation operation with integer exponents only requires elementary algebra.### Positive integer exponents

a2 = a·a is called the square of a because the area of a square with side-length a is a2.a3 = a·a·a is called the cube,
because the volume of a cube with side-length a is a3.

So 32 is pronounced "three squared",and 23 is
"two cubed".

The exponent says 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 repeated multiplication, because the
exponent is 5. Here, 3 is the base, 5 is the exponent, and 243 is
the power or, more specifically, the fifth power of 3 or 3 raised
to the fifth power.

The word "raised" is usually omitted, and most
often "power" as well, so 35 is typically pronounced "three to the
fifth" or "three to the five".

Formally, powers with positive integer exponents
may be defined by the initial condition a1 = a
and the recurrence
relation an+1 = a·an.

### Exponents one and zero

Notice that 31 is the product of only one 3, which is evidently 3.Also note that 35 = 3·34. Also 34 = 3·33.
Continuing this trend, we should have

- 31 = 3·30.

- \frac = x^.

- 1 = \frac = x^ = x^0

Therefore we define 30 = 1 so that the above
equality holds. This leads to the following rule:

- Any number to the power 1 is itself.
- Any nonzero number to the power 0 is 1; one interpretation of these powers is as empty products. The case of 00 is discussed below.

### Combinatorial interpretation

For non-negative integers n and m, the power nm
equals the cardinality of the set of
m-tuples from an n-element
set, or the number of
m-letter words from an n-letter alphabet.

- 05 = | | = 0. There is no 5-tuple from the empty set.
- 14 = | | = 1. There is one 4-tuple from a one-element set.
- 23 = | | = 8. There are 8 3-tuples from a two-element set.
- 32 = | | = 9. There are 9 2-tuples from a three-element set.
- 41 = | | = 4. There are 4 1-tuples from a four-element set.
- 50 = | | = 1. There is exactly one empty tuple.

See also
exponentiation over sets.

### Negative integer exponents

Raising a nonzero number to the −1 power produces its reciprocal.- a^ = \frac

Thus:

- a^ = (a^n)^ = \frac

A negative integer exponent can also be seen as
repeated division
by the base. Thus 3^ = (((1/3)/3)/3)/3 = \frac = \frac.

### Identities and properties

The most important identity satisfied by integer exponentiation is:- a^ = a^m \cdot a^n

- a^ =\frac

- (a^m)^n = a^ .

- (a \cdot b)^n = a^n \cdot b^n.

While addition and multiplication are commutative (for example,
2+3 = 5 = 3+2 and 2·3 = 6 = 3·2), exponentiation is not
commutative: 23 = 8, but 32 = 9.

Similarly, while addition and multiplication are
associative (for
example, (2+3)+4 = 9 = 2+(3+4) and (2·3)·4 = 24 = 2·(3·4),
exponentiation is not associative either: 23 to the 4th power is 84
or 4096, but 2 to the 34 power is 281 or
2,417,851,639,229,258,349,412,352. Without parentheses to modify
the order of calculation, the order is usually understood to be
from right to left:

- a^=a^\ne (a^b)^c=a^=a^

### Powers of ten

Powers of 10 are easily computed in the base ten (decimal) number system. For example, 108 = 100000000.Exponentiation with base 10 is used in
scientific
notation to describe large or small numbers. For instance,
299,792,458 (the speed of
light in a vacuum, in meters per second) can be written as
2.99792458·108 and then approximated as 2.998·108,
(or sometimes as 299.8·106, or 299.8E+6, especially in computer
software).

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 metres.

### Powers of two

The positive powers of 2 are important in computer science because there are 2n possible values for an n-bit variable. See Binary numeral system.Powers of 2 are important in set theory
since a set with n members has a power set, or
set of all subsets of the
original set, with 2n members.

The negative powers of 2 are commonly used, and
the first two have special names: half, and quarter.

### Powers of one

The integer powers of one are one: 1n = 1.### Powers of zero

If the exponent is positive, the power of zero is zero: 0n = 0, where n > 0.If the exponent is negative, the power of zero
(0−n, where n > 0) remains undefined, because division
by zero is implied.

If the exponent is zero, some authors define
00=1, whereas others leave it undefined, as discussed
below.

### Powers of minus one

The powers of minus one are useful for expressing alternating sequences.If the exponent is even, the power of minus one
is one: (−1)2n = 1.

If the exponent is odd, the power of minus one is
minus one: (−1)2n+1 = −1.

### Powers of the imaginary unit

The powers of the imaginary unit i are useful for expressing sequences of period 4. See for example Root of unity#Periodicity.- i^=i \!\

### Powers of e

The number e, the base of the natural logarithm, is a well studied constant approximately equal to 2.718. It can be approximated by large positive or negative powers of numbers close to one, such as- e \approx1.001^\,

- e \approx0.999^\,

- e = \lim_ \left(1+\frac 1 n \right) ^n\,

- e^k = \left(\lim_ \left(1+\frac \right) ^n\right)^k = \lim_ \left(\left(1+\frac \right) ^n\right)^k = \lim_ \left(1+\frac k \right)^ = \lim_ \left(1+\frac k m \right)^m

- e^x =\lim_ \left(1+\frac x n \right)^n

Another popular formula is the power series

- e^x = 1 + x+ \frac2+ \frac6+\cdots+\frac+\cdots \,.

## Powers of real numbers

Raising a positive real number to a power that is
not an integer can be accomplished in two ways.

- Rational number exponents can be defined in terms of nth roots, and arbitrary nonzero exponents can then be defined by continuity.
- The natural logarithm can be used to define real exponents using the exponential function.

The identities
and properties shown above are true for non-integer exponents
as well.

### Principal n-th root

An nth root of a number a is a number b such that bn = a.When referring to the n-th root of a real number a
it is assumed that what is desired is the principal n-th root of
the number. If a is a real number, and n is a positive integer,
then the unique real solution with the same sign as a to the
equation

- \ x^n = a

- x=a^ = \sqrt[n].

For example: 41/2 = 2, 81/3 = 2,
(−8)1/3 = −2.

Note that if n is even,
negative
numbers do not have a principal n-th root.

### Rational powers of positive real numbers

Exponentiation with a rational
exponent m/n can be defined as

- a^ = \left(a^m\right)^ = \sqrt[n].

For example, 82/3 = 4.

Since any real number
can be approximated by rational numbers, exponentiation to an
arbitrary real exponent k can be defined by continuity
with the rule

- a^k = \lim_ a^r,

For example, if

- k \approx 1.732

- 5^k \approx 5^ =5^=\sqrt[250] \approx 16.241 .

### Real powers of positive real numbers

The natural
logarithm ln(x) is the inverse
of the exponential function ex. It is defined for every positive
real number b and satisfies the equation

- b = e^.\,

Assuming bx is already defined, logarithm and
exponent rules give the equality

- b^x = (e^)^x = e^.\,

This equality can be used to define
exponentiation with any positive real base b as

- b^x = e^.\,

This definition of the real number power bx
agrees with the definition given above using rational exponents and
continuity. The definition of exponentiation using logarithms is
more common in the context of complex numbers, as discussed
below.

### Some rational powers of negative real numbers

Neither the logarithm method nor the fractional exponent method can be used to define ak as a real number for a negative real number a and an arbitrary real number k. In some special cases, a definition is possible: integral powers of negative real numbers are real numbers, and rational powers of the form am/n where n is odd can be computed using roots. But since there is no real number x such that x2 = −1, the definition of am/n when n is even and m is odd must use the imaginary unit i, as described more fully in the next section.The logarithm method cannot be used to define ak
as a real number when a x is nonnegative for every real number x,
so log(a) cannot be a real number.

The rational exponent method cannot be used for
negative values of a because it relies on continuity.
The function f(r) = ar has a unique continuous extension from the
rational numbers to the real numbers for each a > 0. But when a
< 0, the function f is not even continuous on the set of
rational numbers r for which it is defined.

For example, take a = −1. The nth root
of −1 is −1 for every odd natural number n. So
if n is an odd positive integer, (−1)(m/n) = −1
if m is odd, and (−1)(m/n) = 1 if m is even. Thus the set
of rational numbers q for which −1q = 1 is dense in the
rational numbers, as is the set of q for which −1q =
−1. This means that the function (−1)q is not
continuous at any rational number q where it is defined.

### Imaginary powers of e

The
geometric interpretation of the operations on complex numbers
and the definition of powers
of e is the clue to understanding e i·x for real x.
Consider the right
triangle (0, 1, 1+i·x/n). For big values of n the
triangle is almost a circular
sector with a small central angle equal to x/n radian. The
triangles (0, (1+i·x/n)k, (1+i·x/n)k+1) are
mutually
similar for all values of k. So for big values of n the
limiting point of (1+ix/n)n is the point on the unit circle
whose angle from the positive real axis is x radians. The polar
coordinates of this point are (r,θ) = (1,x), and the
cartesian
coordinates are (cos(x), sin(x)). So e i·x = cos(x)
+ i·sin(x), and this is Euler's
formula, connecting algebra to trigonometry by means of
complex
numbers.

The solutions to the equation ez = 1 are the
integer multiples of 2·π·i :

- \ = \.

- \ = \ .

More simply: eiπ=−1; ex+iy = ex(cos y+i
sin y) .

### Trigonometric functions

It follows from Euler's formula that the trigonometric functions cosine and sine are- \cos(z) = \frac \qquad \sin(z) = \frac.\,

Historically, cosine and sine were defined
geometrically before the invention of complex numbers. The above
formula reduces the complicated formulas for
trigonometric functions of a sum into the simple exponentiation
formula

- e^=e^\cdot e^.\,

### Complex powers of e

The power ex+i·y is computed ex · ei·y. The real factor ex is the absolute value of ex+i·y and the complex factor ei·y identifies the direction of ex+i·y.### Complex powers of positive real numbers

If a is a positive real number, and z is any
complex number, the power az is defined as ez·ln(a),
where x = ln(a) is the unique real solution to the equation ex = a.
So the same method working for real exponents also works for
complex exponents. For example:

- 2 i = e i·ln(2) = cos(ln(2))+i·sin(ln(2)) = 0.7692+i·0.63896
- e i = 0.54030+i·0.84147
- 10 i = −0.66820+i·0.74398
- (e 2·π) i = 535.49 i = 1

## Powers of complex numbers

Integer powers of complex numbers are defined by
repeated multiplication or division as above. Complex powers of
positive reals are defined via ex as above. These are continuous
functions. Trying to extend these functions to the general case of
non-integer powers of complex numbers that are not positive reals
leads to difficulties. Either we define discontinuous functions or
multivalued
functions. None of these options are entirely
satisfactory.

The rational power of a complex number must be
the solution to an algebraic equation. For example, w = z1/2 must
be a solution to the equation w2 = z. But if w is a solution, then
so is −w, because (−1)2 = 1 . So the algebraic
equation w2 = z is not sufficient for defining z1/2. Choosing one
of the two solutions as the principal value of z1/2 leaves us with
a function that is not continuous, and the usual rules for
manipulating powers lead us astray.

### The logarithm of a complex numbers

One solution, z = log a, to the
equation ez = a, is called the principal
value of the complex logarithm. It is the unique solution whose
imaginary part lies in the interval
(−π, π].

For example, log 1 = 0,
log(−1) = πi, log i = πi/2, and
log(−i) = −πi/2. The principal value of
the logarithm is known as a branch of the logarithm; other branches
can be specified by choosing a different range for the imaginary
part of the logarithm. The boundary between branches is known as a
branch
cut. The principal value has a branch cut extending from the
origin along the negative real axis, and is discontinuous at each
point of the branch cut.

### Complex power of a complex number

The general complex power ab of a nonzero complex
number a is defined as

- a^b = e^ = e^.\,

When the exponent is a rational
number the power z = an/m is a solution to the equation zm = an
.

The computation of complex powers is facilitated
by converting the base a to polar form, as described in detail
below.

### Complex roots of unity

A complex number a such that an = 1 for a
positive integer n is an 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.

If zn = 1 but zk ≠ 1 for all natural
numbers k such that 0 < k < n, then z is called a primitive
nth root of unity. The negative unit −1 is the only
primitive square root of unity. The imaginary
unit i is one of the two primitive 4-th roots of unity; the
other one is −i.

The number e2πi (1/n) is the primitive nth root
of unity with the smallest positive complex
argument. (It is sometimes called the principal nth root of
unity, although this terminology is not universal and should not be
confused with the principal
value of n√1, which is 1.)

The other nth roots of unity are given by

- \left( e^ \right) ^k = e^

### Roots of arbitrary complex numbers

Although there are infinitely many possible values for a general complex logarithm, there are only a finite number of values for the power az in the important special case where z = 1/n and n is a positive integer. These are the nth roots of a; they are solutions of the equation xn = a. As with real roots, a second root is also called a square root and a third root is also called a cube root.It is conventional in mathematics to define a1/n
as the principal value of the root. If a is a positive real number,
it is also conventional to select a positive real number as the
principal value of the root a1/n. For general complex numbers, the
nth root with the smallest argument is often selected as the
principal value of the nth root operation, as with principal values
of roots of unity.

The set of nth roots of a complex number a is
obtained by multiplying the principal value a1/n by each of the nth
roots of unity. For example, the fourth roots of 16 are 2,
−2, 2i, and −2i, because the principal value of
the fourth root of 16 is 2 and the fourth roots of unity are 1,
−1, i, and −i.

### Computing complex powers

It is often easier to compute complex powers by
writing the number to be exponentiated in polar
form. Every complex number z can be written in the polar form

- z = re^ = e^ \,,

In order to compute the complex power ab, write a
in polar form:

- a = r e^ \,.

- \log a = \log r + i \theta \,,

- a^b = e^ = e^. \,

- \left( r^c e^ \right) e^ = \left( r^c e^ \right) \left[ \cos(d \log r + c\theta) + i \sin(d \log r + c\theta) \right].

The following examples use the principal value,
the branch cut which causes θ to be in the interval
(−π, π]. To compute ii, write i in polar
and Cartesian forms:

- i = 1 \cdot e^,\,
- i = 0 + 1i.\,

- \ i^i = \left( 1^0 e^ \right) e^ = e^ \approx 0.2079.

Similarly, to find (−2)3 + 4i, compute
the polar form of −2,

- -2 = 2e^ \, ,

- (-2)^ = \left( 2^3 e^ \right) e^ \approx (2.602 - 1.006 i) \cdot 10^.

The value of a complex power depends on the
branch used. For example, if the polar form i = 1ei(5π/2) is
used to compute i i, the power is found to be
e−5π/2; the principal value of i i, computed
above, is e−π/2.

### Failure of power and logarithm identities

Identities for powers and logarithms that hold
for positive real numbers may fail when the positive real numbers
are replaced by arbitrary complex numbers. There is no method to
define complex powers or the complex logarithm as complex-valued
functions while preserving the identities these operations possess
in the positive real numbers.

An example involving logarithms concerns the rule
log(ab) = b·log a, which holds whenever a is
a positive real number and b is a real number. The following
calculation shows that this identity does not hold in general for
the principal value of the complex logarithm when a is not a
positive real number:

- i\pi = \log(-1) = \log((-i)^2) \neq 2\log(-i) = 2(-i\pi/2) = -i\pi.

An example involving power rules concerns the
identities

- (ab)^c = a^cb^c, \qquad \left ( \frac\right)^c = \frac.

- 1 = (-1\cdot -1)^ \not = (-1)^(-1)^ = -1,

- i = (-1)^ = \left (\frac\right )^ \not = \frac = \frac = -i.

These examples illustrate that complex powers and
logarithms do not behave the same way as their real counterparts,
and so caution is required when working with the complex versions
of these operations.

## Zero to the zero power

The evaluation of 00 presents a problem, because different mathematical reasoning leads to different results. The best choice for its value depends on the context. According to Benson (1999), "The choice whether to define 00 is based on convenience, not on correctness." There are two principal treatments in practice, one from discrete mathematics and the other from analysis.In many settings, especially in foundations and
combinatorics, 00 is defined to be 1. This definition arises in
foundational treatments of the natural numbers as finite
cardinals, and is useful for shortening combinatorial
identities and removing special cases from theorems, as illustrated
below. In many other settings, 00 is left undefined. In calculus, 00 is an indeterminate
form, which must be analyzed rather than evaluated. In general,
mathematical
analysis treats 00 as undefined
in order that the exponential function be continuous.

Justifications for defining 00 = 1 include:

- When 00 is regarded as an empty product of zeros, its value is 1.
- The combinatorial interpretation of 00 is the number of empty tuples of elements from the empty set. There is exactly one empty tuple.
- Equivalently, the set-theoretic interpretation of 00 is the number of functions from the empty set to the empty set. There is exactly one such function, the empty function.
- It greatly simplifies the theory of polynomials and power series
that a constant term can be written ax0 for an arbitrary x. For
example:
- The formula for the coefficients in a product of polynomials would lose much of its simplicity if constant terms had to be treated specially.
- A power series such as \textstyle e^ = \sum_^ \frac is not valid for x = 0 unless 00, which appears in the numerator of the first term of the series, is 1. Otherwise one would need to use the longer identity \textstyle e^ = 1 + \sum_^ \frac .
- The binomial theorem \textstyle(1+x)^n = \sum_^n \binom x^k is not valid for x = 0, unless 00 = 1. By defining 00 to be 1, a special case of the theorem can be eliminated.

- In differential calculus, the power rule \frac x^n = nx^ is not valid for n=1 at x=0 unless 00 = 1. Defining it this way eliminates the need for a special case for the power rule.

In contexts where the exponent may vary
continuously, it is generally best to treat 00 as an ill-defined
quantity. Justifications for treating it as undefined include:

- The value 00 often arises as the formal limit of exponentiated functions, f(x)g(x), when f(x) and g(x) approach 0 as x approaches a (a constant or infinity). There, 00 suggests [lim f(x)]lim g(x), which is a well defined quantity and is the correct value of lim f(x)g(x) when f and g approach nonzero constants, but is not well defined when f and g approach 0. The same reasoning applies to certain powers involving infinity, \infty^0 and 1^\infty. A more abstract way of saying this is the following: The real function xy of the two nonnegative real variables x and y is not continuous at the point (x, y) = (0, 0), and so 00 is not determined by continuity. That is, the function xy has no continuous extension from the open first quadrant to include the point (0,0). The rule in calculus, that \lim_ f(x)^ = (\lim_ f(x))^ whenever both sides of the equation are defined, would fail if 00 were defined.
- The function zz, viewed as a function of a complex number variable z and defined as ez log z is undefined at z = 0 because log z is undefined at z = 0. Moreover, because zz has a logarithmic branch point at z = 0, it is not common to extend the domain of zz to the origin in this context.

### Treatment in programming languages and calculators

The computer programming languages that evaluate 00 to be 1 include bc, Haskell, J, Java, MATLAB, ML, Perl, Python, R, Ruby, Scheme, and SQL. In the .NET Framework, the method System.Math.Pow treats 00 to be 1. Microsoft Excel issues an error when it evaluates 00.Microsoft
Windows' Calculator and Google
search when used for its calculator function evaluate 00 to
1.

Maple
simplifies a0 to 1 and 0a to 0, even if no constraints are placed
on a, and evaluates 00 to 1.

Mathematica
simplifies a0 to 1, even if no constraints are placed on a. It does
not simplify 0a, and it takes 00 to be an indeterminate form.

## Powers with infinity

Exponential expressions involving infinity may be thought of as generalizations of more familiar kinds of exponentiation, but there are at least two sharply distinct kinds of generalization to the infinite case. On the one hand, there is the combinatorial or set theoretic interpretation; see exponentiation of cardinal numbers.On the other hand, one can find expressions such
as ∞0 and 1∞ arising in analysis for the same
reason as 00, and they are undefined for the same reason. That is,
it is true that (lim f(x))lim g(x) = lim f(x)g(x) when f and g
approach nonzero finite constants, but not when they approach 0 or
∞; then, the limit of the power can be anything, not
predictable from the limits of f and g.

However, if you have a number x∞, where
|x|∞ converges to zero.

It does make sense to say that
∞∞ = ∞ if this is simply interpreted
as an abbreviation for the theorem that if f and g both approach
infinity as x approaches a, then lim f(x)g(x) is also infinite.
(Likewise, 7∞ = ∞, (1.3)∞ =
∞, etc.)

## Efficiently computing a power

The simplest method of computing a'n requires n−1 multiplication operations, but it can be computed more efficiently as illustrated by the following example. To compute 2100, note that 100 = 96 + 4 and 96 = 3*32. Compute the following in order:- 22 = 4
- (22)2 = 24 = 16
- (24)2 = 28 = 256
- (28)2 = 216 = 65,536
- (216)2 = 232 = 4,294,967,296
- 232 232 23224 = 2100

In general, the number of multiplication
operations required to compute an can be reduced to Θ(log
n) by using exponentiation
by squaring or (more generally)
addition-chain exponentiation. Finding the minimal sequence of
multiplications (the minimal-length addition chain for the
exponent) for an is a difficult problem for which no efficient
algorithms are currently known, but many reasonably efficient
heuristic algorithms are available.

## Exponential notation for function names

Placing an integer superscript after the name or symbol of a function, as if the function were being raised to a power, commonly refers to repeated function composition rather than repeated multiplication. Thus f3(x) may mean f(f(f(x))); in particular, f -1(x) usually denotes the inverse function of f.However, for historical reasons, a special syntax
applies to the trigonometric
functions: a positive exponent applied to the function's
abbreviation means that the result is raised to that power, while
an exponent of −1 denotes the inverse function. That is,
sin2x is just a shorthand way to write (sin x)2 without
using parentheses, whereas sin−1x refers to the inverse
function of the sine, also
called arcsin x. There is no need for a shorthand for the
reciprocals of trigonometric functions since each has its own name
and abbreviation, for example 1 / sin(x) =
(sin x)−1 is csc x. A similar
convention applies to logarithms, where log2(x) = (log (x))2 and
there is no common abbreviation for log(log(x)).

## Generalizations of exponentiation

### Exponentiation in abstract algebra

Exponentiation for integer exponents can be defined for quite general structures in abstract algebra.Let X be a set with a power-associative
binary
operation, which we will write multiplicatively. In this very
general situation, we can define xn for any element x of X and any
nonzero natural
number n, by simply multiplying x by itself n times; by
definition, power
associativity means that it doesn't matter in which order we
perform the multiplications.

Now additionally suppose that the operation has
an identity
element 1. Then we can define x0 to be equal to 1 for any x.
Now xn is defined for any natural number n, including 0.

Finally, suppose that the operation has inverses,
and that the multiplication is associative (so that the magma is
a group).
Then we can define x−n to be the inverse of xn when n is
a natural number. Now xn is defined for any integer n and any x in
the group.

Exponentiation in this purely algebraic sense
satisfies the following laws (whenever both sides are defined):

- \ x^=x^mx^n
- \ x^=x^m/x^n
- \ x^=1/x^n
- \ x^0=1
- \ x^1=x
- \ x^=1/x
- \ (x^m)^n=x^

If in addition the multiplication operation is
commutative (so that
the set X is an abelian
group), then we have some additional laws:

- (xy)n = xnyn
- (x/y)n = xn/yn

If we take this whole theory of exponentiation in
an algebraic context but write the binary operation additively,
then "exponentiation is repeated multiplication" can be
reinterpreted as "multiplication is
repeated addition".
Thus, each of the laws of exponentiation above has an analogue among laws of
multiplication.

When one has several operations around, any of
which might be repeated using exponentiation, it is common to
indicate which operation is being repeated by placing its symbol in
the superscript. Thus, x*n is x * ··· * x, while x#n is x # ··· #
x, whatever the operations * and # might be.

Superscript notation is also used, especially in
group
theory, to indicate conjugation.
That is, gh = h−1gh, where g and h are elements of some
group.
Although conjugation obeys some of the same laws as exponentiation,
it is not an example of repeated multiplication in any sense. A
quandle is an algebraic
structure in which these laws of conjugation play a central
role.

### Exponentiation over sets

If n is a natural number and A is an arbitrary
set, the expression An is often used to denote the set of ordered
n-tuples of elements of A. This is equivalent to letting An denote
the set of functions from the set to the set A; the n-tuple
(a0, a1, a2, ..., an−1)
represents the function that sends i to ai.

For an infinite cardinal
number κ and a set A, the notation Aκ is
also used to denote the set of all functions from a set of size
κ to A. This is sometimes written κA to
distinguish it from cardinal exponentiation, defined below.

This generalized exponential can also be defined
for operations on sets or for sets with extra structure.
For example, in linear
algebra, it makes sense to index direct sums of
vector
spaces over arbitrary index sets. That is, we can speak of

- \bigoplus_ V_,

If the base of the exponentiation operation is a
set, the exponentiation operation is the Cartesian
product unless otherwise stated. Since multiple Cartesian
products produce an n-tuple, which can be represented by
a function on a set of appropriate cardinality, SN becomes simply
the set of all functions
from N to S in this case:

- S^N \equiv \.\,

### Exponentiation in category theory

In a Cartesian closed category, the exponential operation can be used to raise an arbitrary object to the power of another object. This generalizes the Cartesian product in the category of sets.### Exponentiation of cardinal and ordinal numbers

In set theory, there are exponential operations for cardinal and ordinal numbers.If κ and λ are cardinal
numbers, the expression κλ represents the
cardinality of the set of functions from any set of cardinality
λ to any set of cardinality κ. If κ
and λ are finite then this agrees with the ordinary
exponential operation. For example, the set of 3-tuples of elements
from a 2-element set has cardinality 8.

Exponentiation of cardinal numbers is distinct
from exponentiation of ordinal
numbers, which is defined by a limit
process. In the ordinal numbers, exponentiation is defined by
transfinite
induction. For ordinals α and β, the
exponential αβ is the supremum of the ordinal
product αγα over all γ <
β.

## Repeated exponentiation

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 tetration. Iterating tetration leads to another operation, and so on. This sequence of operations is captured by the Ackermann function and Knuth's up-arrow notation.## Exponentiation in programming languages

The superscript notation xy is convenient in handwriting but inconvenient for typewriters and computer terminals that align the baselines of all characters on each line. Many programming languages have alternate ways of expressing exponentiation that do not use superscripts:- x ↑ y: Algol, Commodore BASIC
- x ^ y: BASIC, J, Matlab, R, Microsoft Excel, TeX (and its derivatives), Haskell (for integer exponents), and most computer algebra systems
- x ** y: Ada, Bash, Fortran, FoxPro, Perl, Python, Ruby, SAS, ABAP, Haskell (for floating-point exponents), Turing
- x * y: APL
- Power(x, y): Microsoft Excel, Delphi/Pascal (declared in "Math"-unit)
- pow(x, y): C, C++, PHP
- Math.pow(x, y): Java, JavaScript, Modula-3
- Math.Pow(x, y): C#
- (expt x y): Common Lisp, Scheme

In Bash, C, C++, C#, Java, JavaScript, PHP,
Python and Ruby, the symbol ^ represents bitwise XOR. In Pascal, it
represents indirection.

## History of the notation

The term power was used by Euclid for the square of a line. Nicolas Chuquet used a form of exponential notation in the 15th century, which was later used by Henricus Grammateus and Michael Stifel. Samuel Jeake introduced the term indices in 1696. In the 16th century Robert Recorde used the terms square, cube, zenzizenzic (fourth power), surfolide (fifth), zenzicube (sixth), second surfolide (seventh) and Zenzizenzizenzic (eighth). Biquadrate has been used to refer to the fourth power as well.Another historical synonym, involution, is now
rare and should not be confused with its more common
meaning.

## References

## See also

## External links

- sci.math FAQ: What is 00?
- Laws of Exponents with derivation and examples
- 1058 Powers of Two

exponentiation in Arabic: أس

exponentiation in Catalan: Potència
aritmètica

exponentiation in Czech: Umocňování

exponentiation in Danish: Potens
(matematik)

exponentiation in German: Potenz
(Mathematik)

exponentiation in Estonian: Astendamine

exponentiation in Spanish: Potenciación

exponentiation in Esperanto: Potenco
(matematiko)

exponentiation in Persian: توان (ریاضی)

exponentiation in French: Exposant
(mathématiques)

exponentiation in Korean: 거듭제곱

exponentiation in Indonesian: Eksponen

exponentiation in Icelandic: Veldi
(stærðfræði)

exponentiation in Italian: Potenza
(matematica)

exponentiation in Hebrew: חזקה (מתמטיקה)

exponentiation in Latin: Potentia
(mathematica)

exponentiation in Hungarian: Hatvány

exponentiation in Dutch: Machtsverheffen

exponentiation in Japanese: 冪乗

exponentiation in Norwegian: Potens
(matematikk)

exponentiation in Polish: Potęga

exponentiation in Portuguese:
Exponenciação

exponentiation in Quechua: Yupa huqariy

exponentiation in Russian: Возведение в
степень

exponentiation in Simple English:
Exponentiation

exponentiation in Slovak: Mocnina

exponentiation in Slovenian: Potenciranje

exponentiation in Finnish: Potenssi

exponentiation in Swedish: Potens
(matematik)

exponentiation in Tagalog: Eksponente

exponentiation in Thai: การยกกำลัง

exponentiation in Vietnamese: Lũy thừa

exponentiation in Yiddish: מדריגה
(מאטעמאטיק)

exponentiation in Chinese:
冪