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Exponentiation

In mathematics, exponentiation is a process generalized from repeated multiplication, in much the same way that multiplication is a process generalized from repeated addition. (The next operation after exponentiation is sometimes called tetration; repeating this process leads to the Ackermann function. )

The simplest case involves a positive integer exponent: For example, 35 = 3 × 3 × 3 × 3 × 3 = 243. Here, 3 is the base, 5 is the exponent (written as a superscript), and 243 is 3 raised to the 5th power or 3 raised to the power 5. (The word "raised" is usually omitted, and most often "power" as well, so 35 is typically pronounced "three to the fifth" or "three to the five".) Notice that the base 3 appears 5 times in the repeated multiplication, because the exponent is 5. In contexts where superscripts are not available, such as computer languages and e-mail, 35 is commonly written "3^5" (with a caret), and sometimes as "3**5" (with two asterisks). Another way of writing it, and a good method available nowadays if you have access to Unicode encoding, is "3↑5" (with an up-arrow; HTML &#8593;).

The exponent 1 is not normally written, since any number to the power 1 is itself. The exponents 2 and 3 occur so commonly that there are short words for them: the powers are called the square and cube of the base, respectively. 32 is pronounced "three squared," and 33 is "three cubed."

The meaning of 35 may also be viewed as 1 × 3 × 3 × 3 × 3 × 3: the starting value 1 (the identity element of multiplication) is multiplied by the base, as many times as indicated by the exponent. With this definition in mind, it is easy to see how to generalize exponentiation to zero and negative exponents: any number to the 0 power is 1, and a negative exponent indicates repeated division by the base. Thus 3-5 = 1 ÷ 3 ÷ 3 ÷ 3 ÷ 3 ÷ 3 = 1/243, and raising any nonzero number to the -1 power produces its reciprocal.

Raising 0 to a negative power would imply division by 0, and so is undefined. 00 is sometimes taken as undefined, but is sensibly defined as 1; see the reference below.

Important identities satisfied by exponentiation include:

• xm+n = xmxn
• xm-n = xm/xn
• (xm)n = xmn

Whereas addition or multiplication are commutative (for example, 2+3 = 5 = 3+2 and 2×3 = 6 = 3×2), this is not true of exponentiation: 23 = 8 while 32 = 9. Similarly, whereas addition or multiplication are associative (for example, (2+3)+4 = 9 = 2+(3+4) and (2×3)×4 = 24 = 2×(3×4)), this is not true of exponentiation either: 23 to the 4th power is 84 or 4,096, while 2 to the 34 power is 281 or 2,417,851,639,229,258,349,412,352.

Powers of 10 are easy to compute: for example 106 = 1 million, which is 1 followed by 6 zeros. Exponentiation with base 10 is often used in the physical sciences to describe large or small numbers in scientific notation; for example, 299792458 (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 if this is useful. SI prefixes are also used to describe small or large quantities, and these are also based on powers of 10; for example, the prefix kilo means 103 = 1000, so a kilometre is 1000 metres.

Powers of 2 are important in computer science; for example, there are 2n possible values for a variable that takes n bits to store in memory. They occur so commonly that SI prefixes are commonly reinterpreted to refer to them: 1 kilobyte = 210 = 1024 bytes. As the standard meanings of the prefixes also occur, confusion may result, and in 1998 the International Electrotechnical Commission approved a set of binary prefixes. For instance, the prefix for multiples of 1024 is kibi-, so 1024 bytes is 1 kibibyte. Other prefixes are mebi-, gibi-, and tebi-.

Exponentiation with a fractional exponent is defined as

$x^{m/n} = \sqrt[n]{x^{m}}.$

So for example 82/3 = 4, and the 1/2 power of a non-negative number is its square root (positive or negative).

Exponentiation to an arbitrary real exponent can then be defined by continuity.

With real numbers, the exponential function exp is the same as raising the transcendental number e to the indicated power: exp x = ex.

Exponentiation of real numbers, and even complex numbers, can 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).

However, in the complex number field, it should be noted that logarithms are always multi-valued functions, usually with an imaginary periodicity. Therefore, expressions such as xy are always similarly multi-valued. A "branch cut" may be created in the complex plane, if a single-valued logarithm or power is desired.

Most often, this branch cut is made along the negative real axis. The use of ex in this context is usually assumed to use this "principle branch" of the logarithm, so the results correspond with that of the exponential function which satisfies analyticity constraints.

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.

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Exponents on function names

When the name or symbol of a function is given an integer superscript, as if being raised to a power, this commonly refers to repeated function composition rather than repeated multiplication. Thus f3(x) may mean f(f(f(x))); in particular, f -1(x) usually denotes f's inverse function.

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 indicates the inverse function. That is, sin2x is just a shorthand way to write (sin x)2 without using parentheses, whereas sin-1x refers to the inverse function of the sine, also called arcsin x. There is no need for a shorthand of this kind for reciprocal trigonometric functions since they each have their own name and abbreviation already: (sin x)-1 is normally just written as csc x.

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, and that the multiplication is associative (so that the magma is a group). 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.

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 multiplication operation is commutative (so that the magma is an abelian group), then we have some additional laws:

• (xy)n = xnyn
• (x/y)n = xn/yn

Notice that in this algebraic context, 00 is always equal to 1. When 00 is attained as a limit, however, 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 is 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 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.

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 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.

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

$\prod_{i \in I} k_{i}$

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

$\bigoplus_{i \in I} V_{i},$

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 exponentiation of cardinal numbers once gain, in the sense that |SI| = |S||I|, where |X| is the cardinality of X. When I=2={0,1}, we have |2X| = 2|X|, where 2X, usually denoted by 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.

Table

Table of kn, with k on the left and n at the top.

 1 2 3 4 5 6 7 8 9 10 1 1 1 1 1 1 1 1 1 1 1 2 2 4 8 16 32 64 128 256 512 1,024 3 3 9 27 81 243 729 2,187 6,561 19,683 59,049 4 4 16 64 256 1,024 4,096 16,384 65,536 262,144 1,048,576 5 5 25 125 625 3,125 15,625 78,125 390,625 1,953,125 9,765,625 6 6 36 216 1,296 7,776 46,656 279,936 1,679,616 10,077,696 60,466,176 7 7 49 343 2,401 16,807 117,649 823,543 5,764,801 40,353,607 282,475,249 8 8 64 512 4,096 32,768 262,144 2,097,152 16,777,216 134,217,728 1,073,741,824 9 9 81 729 6,561 59,049 531,441 4,782,969 43,046,721 387,420,489 3,486,784,401 10 10 100 1,000 10,000 100,000 1,000,000 10,000,000 100,000,000 1,000,000,000 10,000,000,000