 Exponentiation

"Exponent" redirects here. For other uses, see Exponent (disambiguation).
Exponentiation is a mathematical operation, written as a^{n}, involving two numbers, the base a and the exponent (or power) n. When n is a positive integer, exponentiation corresponds to repeated multiplication; in other words, a product of n factors of a (the product itself can also be called power):
just as multiplication by a positive integer corresponds to repeated addition:
The exponent is usually shown as a superscript to the right of the base. The exponentiation a^{n} can be read as: a raised to the nth power, a raised to the power [of] n, or possibly a raised to the exponent [of] n, or more briefly as a to the n. Some exponents have their own pronunciation: for example, a^{2} is usually read as a squared and a^{3} as a cubed. When superscripts cannot be used, as in plain ASCII text, common alternative formats include a^n and a**n.
The power a^{n} can be defined also when n is a negative integer, for nonzero a. No natural extension to all real a and n exists, but when the base a is a positive real number, a^{n} can be defined for all real and even complex exponents n via the exponential function e^{z}. Trigonometric functions can be expressed in terms of complex exponentiation.
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.
Terminology
When this article refers to 'an odd power' of a number it means the exponent is an odd number, not that the result is odd. For instance 2^{3} which is 8 is an odd power of 2 because the exponent is 3. This is the usual usage and applies to any similar form like an even power, negative power, or positive power.
Integer exponents
The exponentiation operation with integer exponents requires only elementary algebra.
Positive integer exponents
The expression a^{2} = a·a is called the square of a because the area of a square with sidelength a is a^{2}.
The expression a^{3} = a·a·a is called the cube, because the volume of a cube with sidelength a is a^{3}.
So 3^{2} is pronounced "three squared", and 2^{3} is "two cubed".
The exponent says how many copies of the base are multiplied together. For example, 3^{5} = 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, 3 raised to the fifth power, or 3 to the power of 5.
The word "raised" is usually omitted, and very often "power" as well, so 3^{5} 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 a^{1} = a and the recurrence relation a^{n+1} = a·a^{n}.
Exponents one and zero
Notice that a^{1} is the "product" of only one a, which is defined to be a. Also note that a^{n − 1} = a^{n}/a. Assuming n = 1, we get a^{0} = 1. Another way of saying this is that when n, m, and n − m are positive (and if a is not equal to zero), one can see that
Extended to the special case when n and m are equal, the equality would read
since both the numerator and the denominator are equal. Therefore we take this as the definition of a^{0}. This leads to the following rule:
 Any number raised to the power 1 is the number itself.
 Any nonzero number raised to the power 0 is 1; one interpretation of these powers is as empty products. The case of 0^{0} is discussed below.
Combinatorial interpretation
For nonnegative integers n and m, the power n^{m} equals the cardinality of the set of mtuples from an nelement set, or the number of mletter words from an nletter alphabet.

0^{5} = │ {} │ = 0. There is no 5tuple from the empty set. 1^{4} = │ { (1,1,1,1) } │ = 1. There is one 4tuple from a oneelement set. 2^{3} = │ { (1,1,1), (1,1,2), (1,2,1), (1,2,2), (2,1,1), (2,1,2), (2,2,1), (2,2,2) } │ = 8. There are eight 3tuples from a twoelement set. 3^{2} = │ { (1,1), (1,2), (1,3), (2,1), (2,2), (2,3), (3,1), (3,2), (3,3) } │ = 9. There are nine 2tuples from a threeelement set. 4^{1} = │ { (1), (2), (3), (4) } │ = 4. There are four 1tuples from a fourelement set. 5^{0} = │ { () } │ = 1. There is exactly one empty tuple.
See also exponentiation over sets.
Negative integer exponents
By definition, raising a nonzero number to the −1 power produces its reciprocal:
One also defines
for any nonzero a and any positive integer n. Raising 0 to a negative power would imply division by 0, so it is left undefined.
The definition of a^{−n} for nonzero a is made so that the identity a^{m}a^{n} = a^{m+n}, initially true only for nonnegative integers m and n, holds for arbitrary integers m and n. In particular, requiring this identity for m = −n is requiring
where a^{0} is defined above, and this motivates the definition a^{−n} = ^{1}/_{an} shown above.
Exponentiation to a negative integer power can alternatively be seen as repeated division of 1 by the base. For instance,
 .
Identities and properties
The most important identity satisfied by integer exponentiation is
This identity has the consequence
for a ≠ 0, and
Another basic identity is
While addition and multiplication are commutative (for example, 2+3 = 5 = 3+2 and 2·3 = 6 = 3·2), exponentiation is not commutative: 2^{3} = 8, but 3^{2} = 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: 2^{3} to the 4th power is 8^{4} or 4096, but 2 to the 3^{4} power is 2^{81} 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 topdown, not bottomup:
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, 10^{3} = 1000 and 10^{−4} = 0.0001.
Exponentiation with base 10 is used in scientific notation to describe large or small numbers. For instance, 299,792,458 meters/second (the speed of light in a vacuum, in meters per second) can be written as 2.99792458·10^{8} m/s and then approximated as 2.998·10^{8} m/s.
SI prefixes based on powers of 10 are also used to describe small or large quantities. For example, the prefix kilo means 10^{3} = 1000, so a kilometre is 1000 metres.
Powers of two
The positive powers of 2 are important in computer science because there are 2^{n} possible values for an nbit binary variable.
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 2^{n} members.
The negative powers of 2 are commonly used, and the first two have special names: half, and quarter.
In the base 2 (binary) number system, integer powers of 2 are written as 1 followed or preceded by a number of zeroes determined by the sign and magnitude of the exponent. For example, two to the power of three is written 1000 in binary.
Powers of 1
The integer powers of one are one: 1^{n} = 1.
Powers of zero
If the exponent is positive, the power of zero is zero: 0^{n} = 0, where n > 0.
If the exponent is negative, the power of zero (0^{n}, where n < 0) is undefined, because division by zero is implied.
If the exponent is zero, some authors define 0^{0}=1, whereas others leave it undefined, as discussed below.
Powers of minus one
If n is an even integer, then (−1)^{n} = 1.
If n is an odd integer, then (−1)^{n} = −1.
Because of this, powers of −1 are useful for expressing alternating sequences. For a similar discussion of powers of the complex number i, see the section on Powers of complex numbers.
Large exponents
The limit of a sequence of powers of a number greater than one diverges, in other words they grow without bound:
 a^{n} → ∞ as n → ∞ when a > 1 .
This can be read as "a to the power of n tends to +∞ as n tends to infinity when a is greater than one".
Powers of a number with absolute value less than one tend to zero:
 a^{n} → 0 as n → ∞ when a < 1 .
Any power of one is always itself:
 a^{n} = 1 for all n if a = 1 .
If the number a varies 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 the section below Powers of e.
Other limits, in particular of those tending to indeterminate forms, are described in limits of powers below.
Real powers of positive 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 n^{th} 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 for integer exponents are true for positive real numbers with noninteger exponents as well. However the identity
cannot be extended consistently to where a is a negative real number, see negative nth roots. The failure of this identity is the basis for the problems with complex number powers detailed under failure of power and logarithm identities.
Principal nth root
Main article: nth rootAn nth root of a number a is a number x such that x^{n} = a.
If a is a positive real number and n is a positive integer, then there is exactly one positive real solution to x^{n} = a. This solution is called the principal nth root of a. It is denoted ^{n}√a, where √ is the radical symbol; alternatively, it may be written a^{1/n}. For example: 4^{1/2} = 2, 8^{1/3} = 2,
When one speaks of the nth root of a positive real number a, one usually means the principal nth root.
Rational exponents
A power of a positive real number a with a rational exponent m/n in lowest terms satisfies
where m is an integer and n is a positive integer.
Powers of e
Main article: Exponential functionThe important mathematical constant e, sometimes called Euler's number, is approximately equal to 2.718 and is the base of the natural logarithm. It provides a path for defining exponentiation with noninteger exponents. It is defined as the following limit where the power goes to infinity as the base tends to one:
The exponential function, defined by
has the x written as a power as it satisfies the basic exponential identity
The exponential function is defined for all integer, fractional, real, and complex values of x. It can even be used to extend exponentiation to some nonnumerical entities such as square matrices; however, the exponential identity only holds when x and y commute.
A short proof that e to a positive integer power k is the same as e^{k} is:
This proof shows also that e^{x+y} satisfies the exponential identity when x and y are positive integers. These results are in fact generally true for all numbers, not just for the positive integers.
Real exponents
Since any real number can be approximated by rational numbers, exponentiation to an arbitrary real exponent x can be defined by continuity with the rule
where the limit as r gets close to x is taken only over rational values of r.
For example, if
then
Exponentiation by a real power is normally accomplished using logarithms instead of using limits of rational powers.
The natural logarithm ln(x) is the inverse of the exponential function e^{x}. It is defined for b > 0, and satisfies
If b^{x} is to preserve the logarithm and exponent rules, then one must have
for each real number x.
This can be used as an alternative definition of the real number power b^{x} and 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.
Negative nth roots
Powers of a positive real number are always positive real numbers. The solution of x^{2} = 4, however, can be either 2 or −2. The principal value of 4^{1/2} is 2, but −2 is also a valid square root. If the definition of exponentiation of real numbers is extended to allow negative results then the result is no longer well behaved.
If n is even, then x^{n} = a has two real solutions if a is positive, which are the positive and negative nth roots. The equation has no solution in real numbers if a is negative.
If n is odd, then x^{n} = a has one real solution. The solution is positive if a is positive and negative if a is negative.
Rational powers m/n, where m/n is in lowest terms, are positive if m is even, negative for negative a if m and n are odd, and can be either sign if a is positive and n is even. (−27)^{1/3} = −3, (−27)^{2/3} = 9, and 4^{3/2} has two roots 8 and −8. Since there is no real number x such that x^{2} = −1, the definition of a^{m/n} when a is negative and n is even must use the imaginary unit i, as described more fully in the section Powers of complex numbers.
Neither the logarithm method nor the rational exponent method can be used to define a^{r} as a real number for a negative real number a and an arbitrary real number r. Indeed, e^{r} is positive for every real number r, so ln(a) is not defined as a real number for a ≤ 0. (On the other hand, arbitrary complex powers of negative numbers a can be defined by choosing a complex logarithm of a.)
The rational exponent method cannot be used for negative values of a because it relies on continuity. The function f(r) = a^{r} 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, consider 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 (−1)^{q} = 1 is dense in the rational numbers, as is the set of q for which (−1)^{q} = −1. This means that the function (−1)^{q} is not continuous at any rational number q where it is defined.
Care needs to be taken when applying the power law identities with negative nth roots. For instance, −27 = (−27)^{((2/3)×(3/2))} = ((−27)^{2/3})^{3/2} = 9^{3/2} = 27 is clearly wrong. The problem here occurs in taking the positive square root rather than the negative one at the last step, but in general the same sorts of problems occur as described for complex numbers in the section Failure of power and logarithm identities.
Complex powers of positive real numbers
Imaginary powers of e
Main article: Exponential functionThe geometric interpretation of the operations on complex numbers and the definition of powers of e is the clue to understanding e^{ix} for real x. Consider the right triangle (0, 1, 1 + ix/n). For big values of n the triangle is almost a circular sector with a small central angle equal to x/n radians. The triangles (0, (1 + ix/n)^{k}, (1 + ix/n)^{k+1}) are mutually similar for all values of k. So for large 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 ^{ix} = cos x + isin x, and this is Euler's formula, connecting algebra to trigonometry by means of complex numbers.
The solutions to the equation e^{z} = 1 are the integer multiples of 2πi:
More generally, if e^{b} = a, then every solution to e^{z} = a can be obtained by adding an integer multiple of 2πi to b:
Thus the complex exponential function is a periodic function with period 2πi.
More simply: e^{iπ} = −1; e^{x + iy} = e^{x}(cos y + i sin y).
Trigonometric functions
Main article: Euler's formulaIt follows from Euler's formula that the trigonometric functions cosine and sine are
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
Using exponentiation with complex exponents may reduce problems in trigonometry to algebra.
Complex powers of e
The power z = e^{x+i·y} can be computed as e^{x} · e^{i·y}. The real factor e^{x} is the absolute value of z and the complex factor e^{i·y} identifies the direction of z.
Complex powers of positive real numbers
If a is a positive real number, and z is any complex number, the power a^{z} is defined as e^{z·ln(a)}, where x = ln(a) is the unique real solution to the equation e^{x} = 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)) ≈ 0.76924 + 0.63896i
 e^{i} ≈ 0.54030 + 0.84147i
 10^{i} ≈ −0.66820 + 0.74398i
 (e^{2π})^{i} ≈ 535.49^{i} ≈ 1
Powers of complex numbers
Integer powers of nonzero complex numbers are defined by repeated multiplication or division as above. If i is the imaginary unit and n is an integer, then i^{n} equals 1, i, −1, or −i, according to whether the integer n is congruent to 0, 1, 2, or 3 modulo 4. Because of this, the powers of i are useful for expressing sequences of period 4.
Complex powers of positive reals are defined via e^{x} as in section Complex powers of positive real numbers above. These are continuous functions.
Trying to extend these functions to the general case of noninteger powers of complex numbers that are not positive reals leads to difficulties. Either we define discontinuous functions or multivalued functions. Neither of these options is entirely satisfactory.
The rational power of a complex number must be the solution to an algebraic equation. Therefore it always has a finite number of possible values. For example, w = z^{1/2} must be a solution to the equation w^{2} = z. But if w is a solution, then so is −w, because (−1)^{2} = 1 . A unique but somewhat arbitrary solution called the principal value can be chosen using a general rule which also applies for nonrational powers.
Complex powers and logarithms are more naturally handled as single valued functions on a Riemann surface. Single valued versions are defined by choosing a sheet. The value has a discontinuity along a branch cut. Choosing one out of many solutions as the principal value leaves us with functions that are not continuous, and the usual rules for manipulating powers can lead us astray.
Any nonrational power of a complex number has an infinite number of possible values because of the multivalued nature of the complex logarithm (see below). The principal value is a single value chosen from these by a rule which, amongst its other properties, ensures powers of complex numbers with a positive real part and zero imaginary part give the same value as for the corresponding real numbers.
Exponentiating a real number to a complex power is formally a different operation from that for the corresponding complex number. However in the common case of a positive real number the principal value is the same.
The powers of negative real numbers are not always defined and are discontinuous even where defined. When dealing with complex numbers the complex number operation is normally used instead.
Complex power of a complex number
For complex numbers a and b with a ≠ 0, the notation a^{b} is ambiguous in the same sense that log a is.
To obtain a value of a^{b}, first choose a logarithm of a; call it log a. Such a choice may be the principal value Log a (the default, if no other specification is given), or perhaps a value given by some other branch of log z fixed in advance. Then, using the complex exponential function one defines
 a^{b} = e^{blog a}
because this agrees with the earlier definition in the case where a is a positive real number and the (real) principal value of log a is used.
If b is an integer, then the value of a^{b} is independent of the choice of log a, and it agrees with the earlier definition of exponentation with an integer exponent.
If b is a rational number m/n in lowest terms with n > 0, then the infinitely many choices of log a yield only n different values for a^{b}; these values are the n complex solutions z to the equation z^{n} = a^{m}.
If b is an irrational number, then the infinitely many choices of log a lead to infinitely many distinct values for a^{b}.
The computation of complex powers is facilitated by converting the base a to polar form, as described in detail below.
A similar construction is employed in quaternions.
Complex roots of unity
Main article: Root of unityA complex number a such that a^{n} = 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 ngon with one vertex on the real number 1.
If z^{n} = 1 but z^{k} ≠ 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 4th roots of unity; the other one is −i.
The number e^{2πi (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.^{[1]})
The other nth roots of unity are given by
for 2 ≤ k ≤ n.
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 a^{z} 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 x^{n} = 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 a^{1/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 a^{1/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 a^{1/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
where r is a nonnegative real number and θ is the (real) argument of z. The polar form has a simple geometric interpretation: if a complex number u + iv is thought of as representing a point (u, v) in the complex plane using Cartesian coordinates, then (r, θ) is the same point in polar coordinates. That is, r is the "radius" r^{2} = u^{2} + v^{2} and θ is the "angle" θ = atan2(v, u). The polar angle θ is ambiguous since any multiple of 2π could be added to θ without changing the location of the point. Each choice of θ gives in general a different possible value of the power. A branch cut can be used to choose a specific value. The principal value (the most common branch cut), corresponds to θ chosen in the interval (−π, π]. For complex numbers with a positive real part and zero imaginary part using the principal value gives the same result as using the corresponding real number.
In order to compute the complex power a^{b}, write a in polar form:
 .
Then
and thus
If b is decomposed as c + di, then the formula for a^{b} can be written more explicitly as
This final formula allows complex powers to be computed easily from decompositions of the base into polar form and the exponent into Cartesian form. It is shown here both in polar form and in Cartesian form (via Euler's identity).
The following examples use the principal value, the branch cut which causes θ to be in the interval (−π, π]. To compute i^{i}, write i in polar and Cartesian forms:
Then the formula above, with r = 1, θ = π/2, c = 0, and d = 1, yields:
Similarly, to find (−2)^{3 + 4i}, compute the polar form of −2,
and use the formula above to compute
The value of a complex power depends on the branch used. For example, if the polar form i = 1e^{i(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}. The set of all possible values for i ^{i} is given by:^{[2]}
So there is an infinity of values which are possible candidates for the value of i^{i}, one for each integer k. All of them have a zero imaginary part so one can say i^{i} has an infinity of valid real values.
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. For example:
 The identity log(a^{b}) = b · log a holds whenever a is a positive real number and b is a real number. But for the principal branch of the complex logarithm one has

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

 The identities (ab)^{c} = a^{c}b^{c} and (a/b)^{c} = a^{c}/b^{c} are valid when a and b are positive real numbers and c is a real number. But a calculation using principal branches shows that

 and
 On the other hand, when c is an integer, the identities are valid for all nonzero complex numbers.
 If exponentiation is considered as a multivalued function then the possible values of (−1×−1)^{1/2} are {1, −1}. The identity holds but saying {1} = {(−1×−1)^{1/2}} is wrong.

 The identity (e^{a})^{b} = e^{ab} holds for real numbers a and b, but assuming its truth for complex numbers leads to the following paradox, discovered in 1827 by Clausen:^{[3]}
 For any integer n, we have:
 but this is false when the integer n is nonzero.
 There are a number of problems in the reasoning:
 The major error is that changing the order of exponentiation in going from line two to three changes what the principal value chosen will be.
 From the multivalued point of view the first error occurs even sooner, it is implicit in the first line and not obvious. It is that e is a real number whereas the result of e^{1+2πin} is a complex number better represented as e+0i. Substituting the complex number for the real on the second line makes the power have multiple possible values. Changing the order of exponentiation from lines two to three also affects how many possible values the result can have.
 For any integer n, we have:
Zero to the zero power
Most authors agree with the statements related to 0^{0} in the two lists below, but make different decisions when it comes to defining 0^{0} or not: see the next subsection.
In most settings not involving continuity in the exponent, interpreting 0^{0} as 1 simplifies formulas and eliminates the need for special cases in theorems. (See the next paragraph for some settings that do involve continuity.) For example:
 Regarding a^{0} as an empty product assigns it the value 1, even when a=0.
 The combinatorial interpretation of 0^{0} is the number of empty tuples of elements from the empty set. There is exactly one empty tuple.
 Equivalently, the settheoretic interpretation of 0^{0} is the number of functions from the empty set to the empty set. There is exactly one such function, the empty function.^{[4]}
 The notation for polynomials and power series rely on defining 0^{0} = 1. Identities like and and the binomial theorem are not valid for x = 0 unless 0^{0} = 1.^{[5]}
 In differential calculus, the power rule is not valid for n = 1 at x = 0 unless 0^{0} = 1.
On the other hand, when 0^{0} arises from a limit of the form , it must be handled as an indeterminate form.
 Limits involving algebraic operations can often be evaluated by replacing subexpressions by their limits; if the resulting expression does not determine the original limit, the expression is known as an indeterminate form.^{[6]} In fact, when f(t) and g(t) are realvalued functions both approaching 0 (as t approaches a real number or ±∞), with f(t) > 0, the function f(t)^{g(t)} need not approach 1; depending on f and g, the limit of f(t)^{g(t)} can be any nonnegative real number or +∞, or it can be undefined. For example, the functions below are of the form f(t)^{g(t)} with f(t),g(t) → 0 as t → 0^{+}, but the limits are different:

 .
 So 0^{0} is an indeterminate form. This behavior shows that the twovariable function x^{y}, though continuous on the set {(x,y): x > 0}, cannot be extended to a continuous function on any set containing (0,0), no matter how 0^{0} is defined.^{[7]} However, under certain conditions, such as when f and g are both analytic functions and f is nonnegative, the limit approaching from the right is always 1.^{[8]}^{[9]}^{[10]}
 In the complex domain, the function z^{w} is defined for nonzero z by choosing a branch of log z and setting z^{w} := e^{w log z}, but there is no branch of log z defined at z = 0, let alone in a neighborhood of 0.^{[11]}
History of differing points of view
Different authors interpret the situation above in different ways:
 Some argue that the best value for 0^{0} depends on context, and hence that defining it once and for all is problematic.^{[12]} According to Benson (1999), "The choice whether to define 0^{0} is based on convenience, not on correctness."^{[13]}
 Others argue that 0^{0} is 1. According to p. 408 of Knuth (1992), it "has to be 1", although he goes on to say that "Cauchy had good reason to consider 0^{0} as an undefined limiting form" and that "in this much stronger sense, the value of 0^{0} is less defined than, say, the value of 0+0" (emphases in original).^{[14]}
The debate has been going on at least since the early 19th century. At that time, most mathematicians agreed that 0^{0} = 1, until in 1821 Cauchy^{[15]} listed 0^{0} along with expressions like ^{0}⁄_{0} in a table of undefined forms. In the 1830s Libri^{[16]}^{[17]} published an unconvincing argument for 0^{0} = 1, and Möbius^{[18]} sided with him, erroneously claiming that whenever A commentator who signed his name simply as "S" provided the counterexample of (e^{−1/t})^{t}, and this quieted the debate for some time, with the apparent conclusion of this episode being that 0^{0} should be undefined. More details can be found in Knuth (1992).^{[14]}
Treatment on computers
IEEE floating point standard
The IEEE 7542008 floating point standard is used in the design of most floating point libraries. It recommends a number of different functions for computing a power:^{[19]}
 pow treats 0^{0} as 1. This is the oldest defined version. If the power is an exact integer the result is the same as for pown, otherwise the result is as for powr (except for some exceptional cases).
 pown treats 0^{0} as 1. The power must be an exact integer. The value is defined for negative bases, e.g. pown(−3,5) is −243.
 powr treats 0^{0} as NaN (NotaNumber  undefined). The value is also NaN for cases like powr(−3,2) where the base is less than zero. The value is defined by e^{power×log(base)}.
Programming languages
Most programming language with a power function are implemented using the IEEE pow function and therefore evaluate 0^{0} as 1. The later C^{[20]} and C++ standards describe this as the normative behaviour. The Java standard^{[21]} mandates this behavior. The .NET Framework method
System.Math.Pow
also treats 0^{0} as 1.^{[22]}Mathematics software
 Sage simplifies a^{0} to 1, even if no constraints are placed on a.^{[23]} It does not simplify 0^{a}, and it takes 0^{0} to be 1.
 Maple simplifies a^{0} to 1 and 0^{a} to 0, even if no constraints are placed on a (the latter simplification is only valid for a > 0), and evaluates 0^{0} to 1.^{[citation needed]}
 Macsyma also simplifies a^{0} to 1 and 0^{a} to 0, even if no constraints are placed on a, but issues an error for 0^{0}.^{[citation needed]}
 Mathematica and Wolfram Alpha simplify a^{0} into 1, even if no constraints are placed on a.^{[24]} While Mathematica does not simplify 0^{a}, Wolfram Alpha returns two results, 0 and indeterminate.^{[25]} Both Mathematica and Wolfram Alpha take 0^{0} to be an indeterminate form.^{[26]}
Limits of powers
The section zero to the zero power gives a number of examples of limits which are of the indeterminate form 0^{0}. The limits in these examples exist, but have different values, showing that the twovariable function x^{y} has no limit at the point (0,0). One may ask at what points this function does have a limit.
More precisely, consider the function f(x,y) = x^{y} defined on D = {(x,y) ∈ R^{2} : x > 0}. Then D can be viewed as a subset of R^{2} (that is, the set of all pairs (x,y) with x,y belonging to the extended real number line R = [−∞, +∞], endowed with the product topology), which will contain the points at which the function f has a limit.
In fact, f has a limit at all accumulation points of D, except for (0,0), (+∞,0), (1,+∞) and (1,−∞).^{[27]} Accordingly, this allows one to define the powers x^{y} by continuity whenever 0 ≤ x ≤ +∞, −∞ ≤ y ≤ +∞, except for 0^{0}, (+∞)^{0}, 1^{+∞} and 1^{−∞}, which remain indeterminate forms.
Under this definition by continuity, we obtain:
 a^{+∞} = +∞ and a^{−∞} = 0, when 1 < a ≤ +∞.
 a^{+∞} = 0 and a^{−∞} = +∞, when 0 ≤ a < 1.
 0^{b} = 0 and (+∞)^{b} = +∞, when 0 < b ≤ +∞.
 0^{b} = +∞ and (+∞)^{b} = 0, when −∞ ≤ b < 0.
These powers are obtained by taking limits of x^{y} for positive values of x. This method does not permit a definition of x^{y} when x < 0, since pairs (x,y) with x < 0 are not accumulation points of D.
On the other hand, when n is an integer, the power x^{n} is already meaningful for all values of x, including negative ones. This may make the definition 0^{n} = +∞ obtained above for negative n problematic when n is odd, since in this case t^{n} → +∞ as t tends to 0 through positive values, but not negative ones.
Efficient computation of integer powers
The simplest method of computing a^{n} requires n−1 multiplication operations, but it can be computed more efficiently than that, as illustrated by the following example. To compute 2^{100}, note that 100 = 64 + 32 + 4. Compute the following in order:
 2^{2} = 4
 (2^{2})^{2} = 2^{4} = 16
 (2^{4})^{2} = 2^{8} = 256
 (2^{8})^{2} = 2^{16} = 65,536
 (2^{16})^{2} = 2^{32} = 4,294,967,296
 (2^{32})^{2} = 2^{64} = 18,446,744,073,709,551,616
 2^{64} 2^{32} 2^{4} = 2^{100} = 1,267,650,600,228,229,401,496,703,205,376
This series of steps only requires 8 multiplication operations instead of 99 (since the last product above takes 2 multiplications).
In general, the number of multiplication operations required to compute a^{n} can be reduced to Θ(log n) by using exponentiation by squaring or (more generally) additionchain exponentiation. Finding the minimal sequence of multiplications (the minimallength addition chain for the exponent) for a^{n} is a difficult problem for which no efficient algorithms are currently known (see Subset sum problem), but many reasonably efficient heuristic algorithms are available.^{[28]}
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 f^{ 3}(x) may mean f(f(f(x))); in particular, f^{ −1}(x) usually denotes the inverse function of f. Iterated functions are of interest in the study of fractals and dynamical systems. Babbage was the first to study the problem of finding a functional square root f^{ 1/2}(x).
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, sin^{2}x is just a shorthand way to write (sin x)^{2} without using parentheses, whereas sin^{−1}x 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} = csc x. A similar convention applies to logarithms, where log^{2}x usually means (log x)^{2}, not log log x.
Generalizations
In abstract algebra
Exponentiation for integer exponents can be defined for quite general structures in abstract algebra.
Let X be a set with a powerassociative binary operation which is written multiplicatively. Then x^{n} is defined for any element x of X and any nonzero natural number n as the product of n copies of x, which is recursively defined by
One has the following properties
 (powerassociative property),
If the operation has a twosided identity element 1 (often denoted by e), then x^{0} is defined to be equal to 1 for any x.
 Two sided identity
If the operation also has twosided inverses, and multiplication is associative then the magma is a group. The inverse of x can be denoted by x^{−1} and follows all the usual rules for exponents.
 Two sided inverse
 Associative
If the multiplication operation is commutative (as for instance in abelian groups), then the following holds:
If the binary operation is written additively, as it often is for abelian groups, 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, g^{h} = h^{−1}gh, 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.
Over sets
Main article: Cartesian productIf n is a natural number and A is an arbitrary set, the expression A^{n} is often used to denote the set of ordered ntuples of elements of A. This is equivalent to letting A^{n} denote the set of functions from the set {0, 1, 2, ..., n−1} to the set A; the ntuple (a_{0}, a_{1}, a_{2}, ..., a_{n−1}) represents the function that sends i to a_{i}.
For an infinite cardinal number κ and a set A, the notation A^{κ} is also used to denote the set of all functions from a set of size κ to A. This is sometimes written ^{κ}A 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
where each V_{i} is a vector space. Then if V_{i} = V for each i, the resulting direct sum can be written in exponential notation as V^{⊕N}, or simply V^{N} with the understanding that the direct sum is the default. We can again replace the set N with a cardinal number n to get V^{n}, although without choosing a specific standard set with cardinality n, this is defined only up to isomorphism. Taking V to be the field R of real numbers (thought of as a vector space over itself) and n to be some natural number, we get the vector space that is most commonly studied in linear algebra, the Euclidean space R^{n}.
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 ntuple, which can be represented by a function on a set of appropriate cardinality, S^{N} becomes simply the set of all functions from N to S in this case:
This fits in with the exponentiation of cardinal numbers, in the sense that S^{N} = S^{N}, where X is the cardinality of X. When "2" is defined as {0,1}, we have 2^{X} = 2^{X}, where 2^{X}, usually denoted by P(X), is the power set of X; each subset Y of X corresponds uniquely to a function on X taking the value 1 for x ∈ Y and 0 for x ∉ Y.
In category theory
Main article: Cartesian closed categoryIn 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. If 0 is an initial object in a Cartesian closed category, then the exponential object 0^{0} is isomorphic to any terminal object 1.
Of cardinal and ordinal numbers
Main articles: cardinal arithmetic and ordinal arithmeticIn set theory, there are exponential operations for cardinal and ordinal numbers.
If κ and λ are cardinal numbers, the expression κ^{λ} represents the cardinality of the set of functions from any set of cardinality λ to any set of cardinality κ.^{[4]} If κ and λ are finite, then this agrees with the ordinary arithmetic exponential operation. For example, the set of 3tuples of elements from a 2element set has cardinality 8 = 2^{3}.
Exponentiation of cardinal numbers is distinct from exponentiation of ordinal numbers, which is defined by a limit process involving transfinite induction.
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 expressed by the Ackermann function and Knuth's uparrow notation. Just as exponentiation grows faster than multiplication, which is faster growing than addition, tetration is faster growing than exponentiation. Evaluated at (3,3), the functions addition, multiplication, exponentiation, tetration yield 6, 9, 27, and 7,625,597,484,987 respectively.
In programming languages
The superscript notation x^{y} 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 BASICx ^ y
: BASIC, J, MATLAB, R, Microsoft Excel, TeX (and its derivatives), TIBASIC, bc (for integer exponents), Haskell (for nonnegative integer exponents), Lua, ASP and most computer algebra systemsx ^^ y
: Haskell (for fractional base, integer exponents), Dx ** y
: Ada, Bash, COBOL, Fortran, FoxPro, Gnuplot, OCaml, Perl, PL/I, Python, Rexx, Ruby, SAS, Tcl, ABAP, Haskell (for floatingpoint exponents), Turing, VHDLx⋆y
: APLPower(x, y)
: Microsoft Excel, Delphi/Pascal (declared in "Math"unit)pow(x, y)
: C, C++, PHP, Tcl, Pythonmath.pow(x, y)
: Python (always fractional results)Math.pow(x, y)
: Java, JavaScript, Modula3, Standard MLMath.Pow(x, y)
orBigInteger.Pow(x, y)
: C# (and other languages using the BCL)(expt x y)
: Common Lisp, Schememath:pow(x, y)
: Erlang
In Bash, C, C++, C#, Java, JavaScript, Perl, PHP, Python and Ruby, the symbol ^ represents bitwise XOR. In Pascal, it represents indirection. In OCaml and Standard ML, it represents string concatenation.
History of the notation
The term power was used by the Greek mathematician Euclid for the square of a line.^{[29]} In the 9th century, Muhammad ibn Mūsā alKhwārizmī used the terms mal for a square and kab for a cube, which later Islamic mathematicians represented in mathematical notation as m and k, respectively, by the 15th century, as seen in the work of Abū alHasan ibn Alī alQalasādī.^{[30]}
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. Samuel Jeake introduced the term indices in 1696.^{[29]} In the 16th century Robert Recorde used the terms square, cube, zenzizenzic (fourth power), surfolide (fifth), zenzicube (sixth), second surfolide (seventh) and Zenzizenzizenzic (eighth).^{[31]} Biquadrate has been used to refer to the fourth power as well.
Some mathematicians (e.g., 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 ax + bxx + cx^{3} + d.
Another historical synonym, involution,^{[32]} is now rare and should not be confused with its more common meaning.
See also
References
 ^ This definition of a principal root of unity can be found in:
 Thomas H. Cormen, Charles E. Leiserson, Ronald L. Rivest, and Clifford Stein (2001). Introduction to Algorithms (second ed.). MIT Press. ISBN 0262032937. Online resource
 Paul Cull, Mary Flahive, and Robby Robson (2005). Difference Equations: From Rabbits to Chaos (Undergraduate Texts in Mathematics ed.). Springer. ISBN 0387232346. Defined on page 351, available on Google books.
 "Principal root of unity", MathWorld.
 ^ Complex number to a complex power may be real at Cut The Knot gives some references to i^{i}
 ^ Steiner J, Clausen T, Abel NH (1827). "Aufgaben und Lehrsätze, erstere aufzulösen, letztere zu beweisen". Journal für die reine und angewandte Mathematik 2: 286–287. http://gdz.sub.unigoettingen.de/no_cache/dms/load/img/?IDDOC=270662.
 ^ ^{a} ^{b} N. Bourbaki, Elements of Mathematics, Theory of Sets, SpringerVerlag, 2004, III.§3.5.
 ^ "Some textbooks leave the quantity 0^{0} undefined, because the functions x^{0} and 0^{x} have different limiting values when x decreases to 0. But this is a mistake. We must define x^{0} = 1, for all x, if the binomial theorem is to be valid when x = 0, y = 0, and/or x = −y. The binomial theorem is too important to be arbitrarily restricted! By contrast, the function 0^{x} is quite unimportant".Ronald Graham, Donald Knuth, and Oren Patashnik (19890105). "Binomial coefficients". Concrete Mathematics (1st ed.). Addison Wesley Longman Publishing Co. p. 162. ISBN 0201142368.
 ^ Malik, S. C.; Savita Arora (1992). Mathematical Analysis. New York: Wiley. p. 223. ISBN 9788122403237. "In general the limit of φ(x)/ψ(x) when x=a in case the limits of both the functions exist is equal to the limit of the numerator divided by the denominator. But what happens when both limits are zero? The division (0/0) then becomes meaningless. A case like this is known as an indeterminate form. Other such forms are ∞/∞ 0×∞, ∞−∞, 0^{0}, 1^{∞} and ∞^{0}."
 ^ L. J. Paige (March 1954). "A note on indeterminate forms". American Mathematical Monthly 61 (3): 189–190. doi:10.2307/2307224. JSTOR 2307224.
 ^ sci.math FAQ: What is 0^0?
 ^ Rotando, Louis M.; Korn, Henry (1977). "The Indeterminate Form 0^{0}". Mathematics Magazine (Mathematical Association of America) 50 (1): 41–42. doi:10.2307/2689754. JSTOR 2689754.
 ^ Lipkin, Leonard J. (2003). "On the Indeterminate Form 0^{0}". The College Mathematics Journal (Mathematical Association of America) 34 (1): 55–56. doi:10.2307/3595845. JSTOR 3595845.
 ^ "... Let's start at x = 0. Here x^{x} is undefined." Mark D. Meyerson, The x^{x} Spindle, Mathematics Magazine 69, no. 3 (June 1996), 198206.
 ^ Examples include Edwards and Penny (1994). Calculus, 4th ed,, PrenticeHall, p. 466, and Keedy, Bittinger, and Smith (1982). Algebra Two. AddisonWesley, p. 32.
 ^ Donald C. Benson, The Moment of Proof : Mathematical Epiphanies. New York Oxford University Press (UK), 1999. ISBN 9780195117219
 ^ ^{a} ^{b} Donald E. Knuth, Two notes on notation, Amer. Math. Monthly 99 no. 5 (May 1992), 403–422.
 ^ AugustinLouis Cauchy, Cours d'Analyse de l'École Royale Polytechnique (1821). In his Oeuvres Complètes, series 2, volume 3.
 ^ Guillaume Libri, Note sur les valeurs de la fonction 0^{0x}, Journal für die reine und angewandte Mathematik 6 (1830), 67–72.
 ^ Guillaume Libri, Mémoire sur les fonctions discontinues, Journal für die reine und angewandte Mathematik 10 (1833), 303–316.
 ^ A. F. Möbius, Beweis der Gleichung 0^{0} = 1, nach J. F. Pfaff, Journal für die reine und angewandte Mathematik 12 (1834), 134–136.
 ^ Handbook of FloatingPoint Arithmetic. Birkhäuser Boston. 2009. p. 216. ISBN 9780817647049.
 ^ John Benito (April 2003) (PDF). Rationale for International Standard—Programming Languages—C. Revision 5.10. p. 182. http://www.openstd.org/jtc1/sc22/wg14/www/C99RationaleV5.10.pdf.
 ^ "Math (Java 2 Platform SE 1.4.2) pow". Oracle. http://download.oracle.com/javase/1.4.2/docs/api/java/lang/Math.html#pow%28double,%20double%29.
 ^ ".NET Framework Class Library Math.Pow Method". Microsoft. http://msdn.microsoft.com/enus/library/system.math.pow.aspx.
 ^ "Sage worksheet calculating x^0". Jason Grout. http://sagenb.org/home/pub/2433/.
 ^ "Wolfram Alpha calculates a^0". Wolfram Alpha LLC, accessed July 24, 2011. http://www.wolframalpha.com/input/?i=a^0.
 ^ "Wolfram Alpha calculates 0^a". Wolfram Alpha LLC, accessed July 24, 2011. http://www.wolframalpha.com/input/?i=0^a.
 ^ "Wolfram Alpha calculates 0^0". Wolfram Alpha LLC, accessed July 24, 2011. http://www.wolframalpha.com/input/?i=0^0.
 ^ N. Bourbaki, Topologie générale, V.4.2.
 ^ Gordon, D. M. 1998. A survey of fast exponentiation methods. J. Algorithms 27, 1 (Apr. 1998), 129146. doi:http://dx.doi.org/10.1006/jagm.1997.0913
 ^ ^{a} ^{b} O'Connor, John J.; Robertson, Edmund F., "Etymology of some common mathematical terms", MacTutor History of Mathematics archive, University of St Andrews, http://wwwhistory.mcs.standrews.ac.uk/Miscellaneous/Mathematical_notation.html.
 ^ O'Connor, John J.; Robertson, Edmund F., "Abu'l Hasan ibn Ali al Qalasadi", MacTutor History of Mathematics archive, University of St Andrews, http://wwwhistory.mcs.standrews.ac.uk/Biographies/AlQalasadi.html.
 ^ Quinion, Michael. "World Wide Words". http://www.worldwidewords.org/weirdwords/wwzen1.htm. Retrieved 20100319
 ^ This definition of "involution" appears in the OED second edition, 1989, and MerriamWebster online dictionary [1]. The most recent usage in this sense cited by the OED is from 1806.
External links
 sci.math FAQ: What is 0^{0}?
 Introducing 0th power on PlanetMath
 Laws of Exponents with derivation and examples
 What does 0^0 (zero to the zeroth power) equal? on AskAMathematician.com
Categories: Exponentials
 Binary operations
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