Exponentiation

In mathematics, exponentiation, denoted b, is an operation involving two numbers: the base, b, and the exponent, n. When n is a positive integer, exponentiation corresponds to repeated multiplication of the base: that is, b is the product of multiplying b by itself n times: \[b^n = \underbrace{b \times b \times \dots \times b \times b}_{n \text{ times}}.\]

The definition of exponentiation can be extended in a natural way to define \(b^x\) for any positive real number base \(b\) and any real number exponent \(x\). A more involved definition supports arbitrary complex numbers as base and exponent. Other generalizations allow algebraic objects, such as square matrices, to be the base.

Exponentiation is a fundamental operation used throughout mathematics, science, and engineering.

This article uses the three common notations for multiplication, \(x\times y\), \(xy\), and \(x\cdot y\), interchangeably.

The exponent is usually shown as a superscript to the right of the base as b. Sometimes an up arrow or caret is used as the operator, e.g. \(b \uparrow n\) or b^n.

In abstract algebra, when there are several operations that may be repeated, it is common to indicate the repeated operation by placing its symbol in the superscript, before the exponent. For example, if f is a function whose valued can be multiplied, \(f^{\circ n}\) denotes exponentiation with respect to function composition, whereas \(f^n\) may denote exponentiation with respect to multiplication. That is,

\((f^n)(x)=(f(x))^n=f(x) \,f(x) \cdots f(x),\)

and

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

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

The exponent is also known as the power, or, in British English, the index. The result of the exponentiation may also be referred to as a power.

The expression b is often read as "b to the power (of) n"; it may also be referred to as "b (raised) to the nth (power)", "the nth power of b", or, most briefly, "b to the n".

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

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

The base case is

\(b^1 = b\)

and the recurrence is

\(b^{n+1} = b^n \cdot b.\)

The associativity of multiplication implies that for any positive integers m and n, when multiplying exponentiation results with the same base b and exponents m and n, the product is the same base b to the power of the sum m + n of the exponents:

\(b^{m+n} = b^m \cdot b^n,\)

and, when taking the result of exponentiation to another power, the base remains the same, and the exponents multiply:

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

These rules can be derived intuitively as follows: \[\begin{align} b^n \times b^m & = \underbrace{b \times \dots \times b}_{n \text{ times}} \times \underbrace{b \times \dots \times b}_{m \text{ times}} \\[1ex] & = \underbrace{b \times \dots \times b}_{n+m \text{ times}} = b^{n+m} \end{align}\] \[\begin{align} (b^m)^n &= \underbrace{\underbrace{b \times \dots \times b}_{m \text{ times}} \times \dots \times \underbrace{b \times \dots \times b}_{m \text{ times}}}_{n \text{ times}} \\[1ex] &= \underbrace{b \times \dots \times b}_{m \times n \text{ times}} = b^{mn} \end{align}\]

A number raised to the 0 power is generally defined to be 1:

\(b^0=1.\)

This ensures that the multiplication rule mentioned above,

\(b^{m+n}=b^m\cdot b^n\)

, also holds for \(m=0\):

\(b^n = 1 \times b^n = b^0 \times b^n = b^{0+n} = b^n\).

This value is also obtained by the empty product convention, which may be used in every algebraic structure with a multiplication that has an identity.

Depending on the context, there may be an exception for zero to the power of zero.

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

\(b^{-n} = \frac{1}{b^n}\)

Raising 0 to a negative exponent, which is related to division by zero, is typically undefined but in some circumstances it may be interpreted as infinity.

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

The same definition applies to invertible elements in a multiplicative monoid, that is, an algebraic structure, with an associative multiplication and a multiplicative identity denoted 1 (for example, the square matrices of a given dimension). In particular, in such a structure, the inverse of an invertible element x is standardly denoted \(x^{-1}.\)

The following identities, often called exponent rules, hold for all integer exponents, provided that the base is not zero:

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

Unlike addition and multiplication, exponentiation is not commutative: for example, \(2^3 = 8\), but reversing the operands gives the different value \(3^2=9\). Also unlike addition and multiplication, exponentiation is not associative: for example, (2) = 8 = 64, whereas 2 = 2 = 512. Without parentheses, the conventional order of operations for serial exponentiation in superscript notation is top-down (or right-associative), not bottom-up (or left-associative). That is,

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

which, in general, is different from

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

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

\((a+b)^n=\sum_{i=0}^n \binom{n}{i}a^ib^{n-i}=\sum_{i=0}^n \frac{n!}{i!(n-i)!}a^ib^{n-i}.\)

However, this formula is true only if the summands commute (i.e. that ab = ba), which is implied if they belong to a structure that is commutative. Otherwise, if a and b are, say, square matrices of the same size, this formula cannot be used. It follows that in computer algebra, many algorithms involving integer exponents must be changed when the exponentiation bases do not commute. Some general purpose computer algebra systems use a different notation (sometimes ^^ instead of ^) for exponentiation with non-commuting bases, which is then called non-commutative exponentiation.

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

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

\(b^n \rightarrow \infty \text{ as } n \rightarrow \infty \text{ when } b > 1\)

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

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

\(b^n \rightarrow 0 \text{ as } n \rightarrow \infty \text{ when } \left|b\right| < 1\)

Any power of one is always one:

\(b^n = 1 \text{ for all } n \text{ for } b = 1\)

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

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

\((1 + \frac{1}{n})^n \rightarrow e \text{ as } n \rightarrow \infty\)

See § Exponential function below.

Condensed: the full section is in Wikipedia.

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

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

For \(c < 0\), the same behavior occurs, but with positive and negative infinity exchanged.

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

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

\(x^\frac{p}{q} = \left(x^p\right)^\frac{1}{q} = \sqrt[q]{x^p} = (x^\frac{1}{q})^p = \left(\sqrt[q]{x}\right)^p.\)

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

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

All these definitions are required for extending the identity \((x^r)^s = x^{rs}\) to rational exponents.

On the other hand, there are problems with the extension of these definitions to bases that are not positive real numbers. For example, a negative real number has a real nth root, which is negative, if n is odd, and no real root if n is even. In the latter case, whichever complex nth root one chooses for ⁠\(x^\frac 1n\)⁠, the identity \((x^a)^b=x^{ab}\) cannot be satisfied. For example,

\(\left((-1)^2\right)^\frac 12 = 1^\frac 12= 1\neq (-1)^{2\cdot\frac 12} =(-1)^1=-1.\)

See § Real exponents and § Non-integer exponents with a complex base for details on the way these problems may be handled.

For positive real numbers, exponentiation to real powers can be defined in two equivalent ways, either by extending the rational powers to reals by continuity (§ Limits of rational exponents, below), or in terms of the logarithm of the base and the exponential function (§ Powers via logarithms, below). The result is always a positive real number, and the identities and properties shown above for integer exponents remain true with these definitions for real exponents. The second definition is more commonly used, since it generalizes straightforwardly to complex exponents.

On the other hand, exponentiation to a real power of a negative real number is much more difficult to define consistently, as it may be non-real and have several values. One may choose one of these values, called the principal value, but there is no choice of the principal value for which the identity

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

is true; see § Failure of power and logarithm identities. Therefore, exponentiation with a basis that is not a positive real number is generally viewed as a multivalued function.

Since any irrational number can be expressed as the limit of a sequence of rational numbers, exponentiation of a positive real number b with an arbitrary real exponent x can be defined by continuity with the rule

\(b^x = \lim_{r (\in \mathbb{Q}) \to x} b^r \quad (b \in \mathbb{R}^+,\, x \in \mathbb{R}),\)

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

For example, if x = π, the non-terminating decimal representation π = 3.14159... and the monotonicity of the rational powers can be used to obtain intervals bounded by rational powers that are as small as desired, and must contain \(b^\pi:\)

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

So, the upper bounds and the lower bounds of the intervals form two sequences that have the same limit, denoted \(b^\pi.\)

This defines \(b^x\) for every positive b and real x as a continuous function of b and x.