A power is an exponent to which a given quantity is raised. The expression is therefore known as " to the th power." A number of powers of are plotted above (cf. Derbyshire 2004, pp. 68 and 73).
The power may be an integer, real number, or complex number. However, the power of a real number to a noninteger power is not necessarily itself a real number. For example, is real only for .
A number other than 0 taken to the power 0 is defined to be 1, which follows from the limit
(1)

This fact is illustrated by the convergence of curves at in the plot above, which shows for , 0.4, ..., 2.0. It can also be seen more intuitively by noting that repeatedly taking the square root of a number gives smaller and smaller numbers that approach one from above, while doing the same with a number between 0 and 1 gives larger and larger numbers that approach one from below. For square roots, the total power taken is , which approaches 0 as is large, giving in the limit that is large.
(zero to the zeroth power) itself is undefined. The lack of a welldefined meaning for this quantity follows from the mutually contradictory facts that is always 1, so should equal 1, but is always 0 (for ), so should equal 0. The choice of definition for is usually defined to be indeterminate, although defining allows some formulas to be expressed simply (Knuth 1992; Knuth 1997, p. 57).
A number to the first power is, by definition, equal to itself, i.e.,
(2)

Similarly,
(3)

for any complex number . It is therefore impressive that Captain Kirk (William Shatner) is able to detect one more heartbeat aboard the starship Enterprise than can be accounted for by amplifying an auditory sensor intensified by a factor of "1 to the fourth power" in the Season 1 Star Trek episode "Court Martial" (1967).
The rules for combining quantities containing powers are called the exponent laws, and the process of raising a base to a given power is known as exponentiation.
The derivative of is given by
(4)

and the indefinite integral by
(5)

The definite integral for real is known as Cavalieri's quadrature formula and is given by
(6)

While the simple equation
(7)

cannot be solved for using traditional elementary functions, the solution can be given in terms of the Lambert Wfunction as
(8)

where is the natural logarithm of .
Similarly, the solution to
(9)

can be solved for in terms of using the Lambert Wfunction. In the special case , in addition to the solutions and , a third solution is
(10)
 
(11)

(OEIS A073084).
Special names given to various powers are listed in the following table.
Expressions of the form are known as power towers.
The largest powers which numbers , 2, 3, ... can be represented in the form are 1, 1, 1, 2, 1, 1, 1, 3, 2, 1, ... (OEIS A052409), with corresponding values of given by 1, 2, 3, 2, 5, 6, 7, 2, 3, 10, ... (OEIS A052410).
A double binomial sum gives the power function as follows,
(12)

(K. MacMillan, pers. comm., Nov. 14, 2007).
The power sum of the first positive integers is given by Faulhaber's formula,
(13)

where is the Kronecker delta, is a binomial coefficient, and is a Bernoulli number.
Let be the largest integer that is not the sum of distinct th powers of positive integers (Guy 1994). The first few values for , 3, ... are 128, 12758, 5134240, 67898771, ... (OEIS A001661).
Catalan's conjecture (now a theorem) states that 8 and 9 ( and ) are the only consecutive powers (excluding 0 and 1), i.e., the only solution to Catalan's Diophantine problem. In addition, Hyyrő and Makowski proved that there do not exist three consecutive powers (Ribenboim 1996).
Very few numbers of the form are prime (where composite powers need not be considered, since ). The only prime numbers of the form for and prime correspond to and the Mersenne primes, i.e., , , , .... Other numbers of the form equal . The only prime numbers of the form for and prime correspond to with , 2, 4, 6, 10, 14, 16, 20, 24, 26, ... (OEIS A005574). Other numbers of the form equal .
There are no nontrivial solutions to the equation
(14)

for (Guy 1994, p. 153).