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|
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 non-integer
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, for 
, agrees with the limit
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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.
The value of 
(zero to the zeroth power) depends on the convention being
used. Defining 
allows many formulas in combinatorics
to hold without exceptions. In limit calculations, however,
is an indeterminate
form because the limit of 
need not have a unique value when 
and 
both approach 0 (Knuth 1992; Knuth 1997, p. 57).
A number to the first power is, by definition, equal to itself, i.e.,
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Similarly,
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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.
For 
, with a fixed analytic branch
of the complex logarithm, the derivative
of 
is given by
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and the indefinite integral by
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For a real number 
and 
,
the definite integral is known as Cavalieri's
quadrature formula and is given by
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While the simple equation
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for 
and 
cannot in general be solved for 
using traditional elementary
functions, the solution can be given in terms of the Lambert
W-function as
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where 
is the natural
logarithm of 
and the appropriate branch of 
is chosen. For 
, the solution is 
.
Similarly, the solution to
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can be solved for 
in terms of 
using the Lambert W-function. In the
special case 
,
in addition to the solutions 
and 
, a third solution is
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(10)
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(11)
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(OEIS A073084).
Special names given to various powers are listed in the following table.
Expressions of the form 
are known as power
towers.
For integers 
, the largest positive
integer powers 
for which 
with an integer 
are 1, 1, 2, 1, 1, 1, 3, 2, 1,
... for 
, 3, 4, ... (OEIS A052409).
The corresponding bases 
are 2, 3, 2, 5, 6, 7, 2, 3, 10, ... (OEIS A052410).
For 
, there is no finite largest exponent,
since 
for every positive
integer 
.
Sequence OEIS A052409 uses 0 at 
by convention.
For a positive integer 
, a double binomial sum gives
the power function as follows,
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(K. MacMillan, pers. comm., Nov. 14, 2007).
For a nonnegative integer 
, the power sum of the first 
positive
integers is given by Faulhaber's formula,
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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 perfect
powers with integer bases
and exponents greater than 1, i.e., the only solution
to Catalan's Diophantine problem.
In addition, Hyyrő and Makowski proved that there do not exist three consecutive
such perfect 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 integers 
and prime 
correspond to 
with 
,
4, 6, 10, 14, 16, 20, 24, 26, ... (OEIS A005574,
which also includes 
).
The case 
gives the prime
2 for every prime 
. For odd prime 
, 
is composite when 
.
There are no nontrivial solutions to the equation
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for integers 
, 
with 
(Guy 1994, p. 153).
and 
" and "The Noninteger Powers 
." Ch. 11 and 13 in