In mathematics, exponentiation, denoted , is an operation involving two numbers: the base, , and the exponent, . When is a positive integer, exponentiation corresponds to repeated multiplication of the base: that is, is the product of multiplying by itself times:
The definition of exponentiation can be extended in a natural way to define for any positive real number base and any real number exponent . 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.
Notation
This article uses the three common notations for multiplication, , , and , interchangeably.
The exponent is usually shown as a superscript to the right of the base as . Sometimes an up arrow or caret is used as the operator, e.g. 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 is a function whose valued can be multiplied, denotes exponentiation with respect to function composition, whereas may denote exponentiation with respect to multiplication. That is,
and
Commonly, is denoted while is denoted
Terminology
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. (In chapter XVI, "Of Powers in general.") Reprinted by Springer-Verlag, 1984; ; ; .
The expression is often read as " to the power (of) "; it may also be referred to as " (raised) to the th (power)", "the th power of ", or, most briefly, " to the ".
The expression is called "the square of " or " squared", because the area of a square with side-length is . Similarly, the expression is called "the cube of " or " cubed", because the volume of a cube with side-length is .
Integer exponents
The exponentiation operation with integer exponents may be defined directly from elementary arithmetic operations.
Positive exponents 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
and the recurrence is
The associativity of multiplication implies that for any positive integers and , when multiplying exponentiation results with the same base and exponents and , the product is the same base to the power of the sum of the exponents:
and, when taking the result of exponentiation to another power, the base remains the same, and the exponents multiply:
These rules can be derived intuitively as follows:
Zero exponent A number raised to the power is generally defined to be :
This ensures that the multiplication rule mentioned above,
, also holds for :
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.
Negative exponents Exponentiation with negative exponents is defined by the following identity, which holds for any integer and nonzero :
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 to negative exponents (consider the case ).
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 (for example, the square matrices of a given dimension). In particular, in such a structure, the inverse of an invertible element is standardly denoted
Identities and properties
The following identities, often called , hold for all integer exponents, provided that the base is not zero:
Unlike addition and multiplication, exponentiation is not commutative: for example, , but reversing the operands gives the different value . Also unlike addition and multiplication, exponentiation is not associative: for example, , whereas . 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,
which, in general, is different from
Powers of a sum The powers of a sum can normally be computed from the powers of the summands by the binomial formula
However, this formula is true only if the summands commute (i.e. that ), which is implied if they belong to a structure that is commutative. Otherwise, if and 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.
Combinatorial interpretation
For nonnegative integers and , the value of is the number of functions from a set of elements to a set of elements (see cardinal exponentiation). Such functions can be represented as -tuples from an -element set (or as -letter words from an -letter alphabet). Some examples for particular values of and are given in the following table:
The possible -tuples of elements from the set 0 = 01 = 1(1, 1, 1, 1)2 = 8(1, 1, 1), (1, 1, 2), (1, 2, 1), (1, 2, 2), (2, 1, 1), (2, 1, 2), (2, 2, 1), (2, 2, 2)3 = 9(1, 1), (1, 2), (1, 3), (2, 1), (2, 2), (2, 3), (3, 1), (3, 2), (3, 3)4 = 4(1), (2), (3), (4)5 = 1()
Particular bases
Powers of ten
In the base ten (decimal) number system, integer powers of are written as the digit followed or preceded by a number of zeroes determined by the sign and magnitude of the exponent. For example, and .
Exponentiation with base is used in scientific notation to denote large or small numbers. For instance, (the speed of light in vacuum, in metres per second) can be written as and then approximated as .
Metric prefixes based on powers of are also used to describe small or large quantities. For example, the prefix kilo- means , so a kilometre is .
Powers of two
The first negative powers of have special names: is a half; is a quarter.
Powers of appear in set theory, since a set with members has a power set, the set of all of its subsets, which has members.
Integer powers of are important in computer science. The positive integer powers give the number of possible values for an -bit integer binary number; for example, a byte may take different values. The binary number system expresses any number as a sum of powers of , and denotes it as a sequence of and , separated by a binary point, where indicates a power of that appears in the sum; the exponent is determined by the place of this : the nonnegative exponents are the rank of the on the left of the point (starting from ), and the negative exponents are determined by the rank on the right of the point.
Powers of one Every power of one equals: .
Powers of zero For a positive exponent , the th power of zero is zero: . For a negative exponent, is undefined.
In some contexts (e.g., combinatorics), the expression is defined to be equal to ; in others (e.g., analysis), it is often undefined.
Powers of negative one Since a negative number times another negative is positive, we have:
Because of this, powers of are useful for expressing alternating sequences. For a similar discussion of powers of the complex number , see .
Large exponents The limit of a sequence of powers of a number greater than one diverges; in other words, the sequence grows without bound:
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:
Any power of one is always one:
Powers of a negative number alternate between positive and negative as alternates between even and odd, and thus do not tend to any limit as grows.
If the exponentiated number varies while tending to as the exponent tends to infinity, then the limit is not necessarily one of those above. A particularly important case is
See below.
Other limits, in particular those of expressions that take on an indeterminate form, are described in below.
Power functions
Real functions of the form , where , are sometimes called power functions. When is an integer and , two primary families exist: for even, and for odd. In general for , when is even will tend towards positive infinity with increasing , and also towards positive infinity with decreasing . All graphs from the family of even power functions have the general shape of , flattening more in the middle as increases. Functions with this kind of symmetry are called even functions.
When is odd, 's asymptotic behavior reverses from positive to negative . For , will also tend towards positive infinity with increasing , but towards negative infinity with decreasing . All graphs from the family of odd power functions have the general shape of , flattening more in the middle as increases and losing all flatness there in the straight line for . Functions with this kind of symmetry are called odd functions.
For , the same behavior occurs, but with positive and negative infinity exchanged.
Table of powers b b2 b3 b4 b5 b6 b7 b8 b9 b10 bn for = OEIS number1 1 1 1 1 1 1 1 1 1 1 2 4 8 16 32 64 128 256 512 3 9 27 81 243 729 4 16 64 256 5 25 125 625 6 36 216 7 49 343 8 64 512 9 81 729 10 100 OEIS number
Rational exponents
If is a nonnegative real number, and is a positive integer, or denotes the unique nonnegative real th root of , that is, the unique nonnegative real number such that
If is a positive real number, and is a rational number, with and integers, then is defined as
The equality on the right may be derived by setting and writing
If is a positive rational number, , by definition.
All these definitions are required for extending the identity 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 th root, which is negative, if is odd, and no real root if is even. In the latter case, whichever complex th root one chooses for , the identity cannot be satisfied. For example,
See and for details on the way these problems may be handled.
Real exponents 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 (, below), or in terms of the logarithm of the base and the exponential function (, 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
is true; see . Therefore, exponentiation with a basis that is not a positive real number is generally viewed as a multivalued function.
Limits of rational exponents
Since any irrational number can be expressed as the limit of a sequence of rational numbers, exponentiation of a positive real number with an arbitrary real exponent can be defined by continuity with the rule