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Raising 10 to a power is easy: for example 10^{7} = 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 × 10^{8} and then approximated[?] as 2.998 × 10^{8} 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 10^{3} = 1000, so a kilometre is 1000 metres.
Exponents with base 2 are used in computer science; for example, there are 2^{n} possible values for a variable that takes n bits to store in memory. A kilobyte usually stands for 2^{10} = 1024 bytes, but sometimes also for 10^{3} = 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 = e^{x}.
We define exponentiation of a positive real number x with a negative exponent by
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
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 powerassociative binary operation, which we will write multiplicatively. In this very general situation, we can define x^{n} 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 x^{0} to be equal to 1 for any x. Now x^{n} 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 x^{n} when n is a natural number. Now x^{n} is defined for any integer n.
In particular, x^{n} 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):
If in addition the operation is commutative and alternative, then we have some additional laws:
Notice that in this algebraic context, 0^{0} is always equal to 1. In some contexts involving calculus, it may be more useful to leave 0^{0} 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 0^{0} be 1, just like every other case of x^{0}. For example, if you expand (0 + x)^{n} using the binomial theorem, you'll want to use 0^{0} = 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, 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'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.
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
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
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, S^{I} 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 S^{I} = 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, a^{b} is the smallest ordinal number greater than a^{c} for c < b when b is a limit ordinal, and of course a^{b+1} := a^{b}a.
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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