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:

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 aⁿ can be read as: a raised to the n-th power, a raised to the power [of] n or possibly a raised to the exponent [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 have their own pronunciation: for example, a² is usually read as a squared and a³ as a cubed.

The power aⁿ 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, aⁿ 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.

Graphs of y=a^x for various bases a: base 10 (green), base e (red), base 2 (blue), and base ½ (cyan). Each curve passes through the point (0,1) because any nonzero number raised to the power 0 is 1. At x=1, the y-value equals the base because any number raised to the power 1 is itself.

[ With integer exponents

The exponentiation operation with integer exponents requires only elementary algebra.

[ Positive integer exponents

The expression a² = a·a is called the square of a because the area of a square with side-length a is a².

The expression 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, 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 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

• 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 nonnegative integers n and m, the power n^m 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 = | { (1,1,1,1) } | = 1. There is one 4-tuple from a one-element set.

2³ = | { (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 3-tuples from a two-element set.

3² = | { (1,1), (1,2), (1,3), (2,1), (2,2), (2,3), (3,1), (3,2), (3,3) } | = 9. There are nine 2-tuples from a three-element set.

4¹ = | { (1), (2), (3), (4) } | = 4. There are four 1-tuples from a four-element set.

5⁰ = | { () } | = 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^maⁿ = 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 a⁰ 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

(a^m)ⁿ = a^mn

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: 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 top-down, not bottom-up:

[ Powers of ten

See Scientific notation

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

Exponentiation with base 10 is used in scientific notation to 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.

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.

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 one

[ Powers of zero

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

[ Powers of minus one

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 the imaginary unit.

[ Large exponents

The limit of a sequence of powers of a number greater than one diverges, in other words they grow without bound:

aⁿ → ∞ as n → ∞ when a > 1 .

This can be read as "a to the power of n tends to infinity as n tends to infinity when a is greater than one".

Powers of a number with absolute value less than one tend to zero:

aⁿ → 0 as n → ∞ when |a| < 1 .

Powers of one, are one:

aⁿ → 1 as n → ∞ when 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+n⁻¹)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 real numbers

• 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 are true for noninteger exponents as well.

[ Principal n-th root

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

An n-th root of a number a is a number x such that xⁿ = a.

If a is a positive real number and n is a positive integer, then there is exactly one positive real solution to xⁿ = a. This solution is called the principal n-th 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 n-th root of a positive real number a, one usually means the principal n-th root.

[ Rational powers

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. (If a were negative, it would be necessary to further restrict m to be an even integer.)

[ Powers of e

The 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 e.g. 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 powers

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 be defined to as to preserve the logarithm and exponent rules, then one must have

This motivates the definition

for each real number x.

This definition of the real number power b^x 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 n-th roots

If n is even, then xⁿ = a has two solutions if a is positive, the positive and negative square roots. The equation has no solution in real numbers if a is negative.

If n is odd, then xⁿ = 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 x² = −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, take a = −1. The n^th 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 n-th 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

The exponential function e^z can be defined as the limit of (1 + z/N)^N, as N approaches infinity, and thus e^iπ is the limit of (1 + iπ/N)^N. In this animation N takes various increasing values from 1 to 100. The computation of (1 + iπ/N)^N is displayed as the combined effect of N repeated multiplications in the complex plane, with the final point being the actual value of (1 + iπ/N)^N. It can be seen that as N gets larger (1 + iπ/N)^N approaches a limit of −1. Therefore, e^iπ = −1, which is known as Euler's identity.

The 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 2iπ:

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

It 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 many problems in trigonometry can be reduced to algebra.

[ Complex powers of e

The power e^x+i·y is computed e^x · e^i·y. The real factor e^x is the absolute value of e^x+i·y and the complex factor e^i·y identifies the direction of e^x+i·y.

[ Complex powers of positive real numbers

[ 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 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 w² = 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 multi-valued 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. For example: (−1)^1/3 = −1 as a real, but when dealing with complex numbers (−1)^1/3 normally means either the principal value e^πi/3 or the set of values {e^πi/3, e^−πi/3, −1}.

[ Powers of the imaginary unit

If i is the imaginary unit and n is an integer, then iⁿ 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 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^bloga

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 n/m in lowest terms with m > 0, then the infinitely many choices of log a yield only m different values for a^b; these values are the m complex solutions z to the equation z^m = aⁿ.

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.

[ 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 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 4-th 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

[ 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" r² = u² + v² 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 thee complex power a^b, write a in polar form:

.

Then

and thus

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