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Exponentiation

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In mathematics, exponentiation is a process of repeated multiplication, in much the same way that multiplication is a process of repeated addition. For example, 34 equals 3 × 3 × 3 × 3 equals 81. Here, 3 is the base, 4 is the exponent (written as a superscript[?]), and 81 is 3 raised to the 4th power. Notice that the base 3 appears 4 times in the repeated multiplication, because the exponent is 4. In contexts where superscripts are not available, such as computer languages and e-mail, 34 is commonly written "3^4".

Raising 10 to a power is easy: for example 107 = 10,000,000 with seven zeros. Exponentiation with base 10 is often used in the physical sciences to describe large or small numbers in scientific notation; for example, 299792458 can be written as 2.99792458 × 108 and then approximated[?] as 2.998 × 108 if this is useful. SI prefixes are also used to describe small or large quantities, but even these are based on powers of ten; for example, the prefix kilo means 103 = 1000, so a kilometre is 1000 metres.

Exponents with base 2 are used in computer science; for example, there are 2n possible values for a variable that takes n bits to store in memory. A kilobyte usually stands for 210 = 1024 bytes, but sometimes also for 103 = 1000 bytes; the term kibibyte has been suggested for the former meaning.

Exponents with base e (a transcendental number approximately equal to 2.71828) are described by the exponential function exp x = ex.

We define exponentiation of a positive real number x with a negative exponent by

x-n = 1/xn
and with a fractional exponent as
<math>x^{m/n} = \sqrt[n]{x^{m}}.</math>
So for instance 10−3 = 0.001 and 82/3 = 4. xy where y is an arbitrary real number can then be defined by continuity.

Exponentiation of real numbers, and even complex numbers, can also be understood with the aid of the exponential function and its inverse, the natural logarithm; in general, we can define

xy = exp (y ln x).

For more on exponents in real and complex numbers, and other situations relevant to mathematical analysis, see Exponential function. That article also lists certain exponential laws (more general than the algebraic laws listed below) that apply in these situations.

Exponentiation in abstract algebra

Exponentiation can also be understood purely in terms of abstract algebra, if we limit the exponents to integers.

Specifically, suppose that X is 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. 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.

In particular, xn is defined for any integer n and any element x of a group. However, because we need only power associativity and not general associativity, the concept of exponentiation also makes sense in some other useful situations, such as the nonzero octonions.

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

  • xm+n = xmxn
  • xm-n = xm/xn
  • x-n = 1/xn = (1/x)n
  • x0 = 1
  • x1 = x
  • x-1 = 1/x
  • (xm)n = xmn
Here, we use a division slash ("/") to indicate multiplying by an inverse, in order to reserve the symbol x-1 for raising x to the power -1, rather than the inverse of x. However, as one of the laws above states, x-1 is always equal to the inverse of x, so the notation doesn't matter in the end.

If in addition the operation is commutative and alternative, then we have some additional laws:

  • (xy)n = xnyn
  • (x/y)n = xn/yn
Here, alternativity is a condition stronger than power associativity but weaker than general associativity. So in particular, this law is satisfied in an Abelian group, such as the multiplicative group of elements from a given field that are distinct from zero.

Notice that in this algebraic context, 00 is always equal to 1. In some contexts involving calculus, it may be more useful to leave 00 undefined.

However, when exponentiation is purely algebraic, that is when the exponents are taken only to be integers, then it's generally most useful to let 00 be 1, just like every other case of x0. For example, if you expand (0 + x)n using the binomial theorem, you'll want to use 00 = 1.

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's 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.

Exponential 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's 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

The above algebraic treatment of exponentiation builds a finitary operation[?] out of a binary operation. In more general contexts, one may be able to define an infinitary operation[?] directly on an indexed set[?].

For example, in the arithmetic of cardinal numbers, it makes sense to say

<math>\prod_{i \in I} k_{i}</math>
for any index set[?] I and cardinal numbers ki. By taking ki = k for every i, this can be interpreted as a repeated product, and the result is kI. In fact, this result depends only on the cardinality of I, so we can define exponentiation of cardinal numbers so that kl is kI for any set I whose cardinality is l.

This can be done even for operations on sets or 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

<math>\bigoplus_{i \in I} V_{i},</math>
where each Vi is a vector space. Then if Vi = V for each i, the resulting direct sum can be written in exponential notation as V(+)I, or simply VI with the understanding that the direct sum is the default. We can again replace the set I with a cardinal number k to get Vk, although without choosing a specific standard set with cardinality k, 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 k to be some natural number n, we get the vector space that is most commonly studied in linear algebra, the Euclidean space Rn.

If the base of the exponentiation operation is itself a set, then by default we assume the operation to be the Cartesian product. In that case, SI becomes simply the set of all functions from I to S. This fits in with the exponentation of cardinal numbers once gain, in the sense that |SI| = |S||I|, where |X| is the cardinality of X. We also have |PX| = 2|X|, where PX is the power set of X. (This is where the term "power set" comes from.)

Note that exponentiation of cardinal numbers doesn't match up with exponentiation of ordinal numbers, which is defined by a limit process. In the ordinal numbers, ab is the smallest ordinal number greater than ac for c < b when b is a limit ordinal, and of course ab+1 := aba.

In category theory, we learn to raise any object in a wide variety of categories to the power of a set, or even to raise an object to the power of an object, using the exponential[?].

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