A power is an exponent to which a given quantity is raised. The expression x^a is therefore known as "x to the ath power." A number of powers of x 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, x^(1/2) is real only for x>=0.


A number other than 0 taken to the power 0 is defined to be 1, which follows from the limit


This fact is illustrated by the convergence of curves at x=0 in the plot above, which shows a^x for a=0.2, 0.4, ..., 2.0. It can also be seen more intuitively by noting that repeatedly taking the square root of a number >1 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 n square roots, the total power taken is 2^(-n), which approaches 0 as n is large, giving a^0=1 in the limit that n is large.

0^0 (zero to the zeroth power) itself is undefined. The lack of a well-defined meaning for this quantity follows from the mutually contradictory facts that a^0 is always 1, so 0^0 should equal 1, but 0^a is always 0 (for a>0), so 0^0 should equal 0. The choice of definition for 0^0 is usually defined to be indeterminate, although defining 0^0=1 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.,




for any complex number z. 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 z^n is given by


and the indefinite integral by

 intz^ndz={(z^(n+1))/(n+1)+C   for n!=-1; lnz+C   for n=-1.

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

 int_a^bx^ndx={(b^(n+1)-a^(n+1))/(n+1)   for n!=-1; ln(b/a)   for n=-1.

While the simple equation


cannot be solved for x using traditional elementary functions, the solution can be given in terms of the Lambert W-function as


where lna is the natural logarithm of a.

Similarly, the solution to


can be solved for a in terms of b using the Lambert W-function. In the special case b=2, in addition to the solutions a=2 and a=4, a third solution is


(OEIS A073084).

Special names given to various powers are listed in the following table.

Expressions of the form x^(x^(·^(·^·))) are known as power towers.

The largest powers p which numbers n=1, 2, 3, ... can be represented in the form n=a^p are 1, 1, 1, 2, 1, 1, 1, 3, 2, 1, ... (OEIS A052409), with corresponding values of a given by 1, 2, 3, 2, 5, 6, 7, 2, 3, 10, ... (OEIS A052410).

A double binomial sum gives the power function as follows,

 sum_(k=1)^nsum_(j=1)^k(-1)^(k-j)j^n(k; j)(x; k)=x^n

(K. MacMillan, pers. comm., Nov. 14, 2007).

The power sum of the first n positive integers is given by Faulhaber's formula,

 sum_(k=1)^nk^p=1/(p+1)sum_(k=1)^(p+1)(-1)^(delta_(kp))(p+1; k)B_(p+1-k)n^k,

where delta_(kp) is the Kronecker delta, (n; k) is a binomial coefficient, and B_k is a Bernoulli number.

Let s_n be the largest integer that is not the sum of distinct nth powers of positive integers (Guy 1994). The first few values for n=2, 3, ... are 128, 12758, 5134240, 67898771, ... (OEIS A001661).

Catalan's conjecture (now a theorem) states that 8 and 9 (2^3 and 3^2) 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 n^p+/-1 are prime (where composite powers p=kb need not be considered, since n^(kb)+/-1=(n^k)^b+/-1). The only prime numbers of the form n^p-1 for n in N and prime p correspond to n=2 and the Mersenne primes, i.e., 2^2-1=3, 2^3-1=7, 2^5-1=31, .... Other numbers of the form n^p-1 equal (n-1)×sum_(k=0)^(p-1)(n)^k. The only prime numbers of the form n^p+1 for n in N and prime p correspond to p=2 with n=1, 2, 4, 6, 10, 14, 16, 20, 24, 26, ... (OEIS A005574). Other numbers of the form n^p+1 equal (n+1)×sum_(k=0)^(p-1)(-n)^k.

There are no nontrivial solutions to the equation


for m<=10^(2000000) (Guy 1994, p. 153).

See also

Apocalyptic Number, Biquadratic Number, Catalan's Conjecture, Catalan's Diophantine Problem, Cavalieri's Quadrature Formula, Circle Power, Complex Exponentiation, Cube Root, Cubed, Cubic Number, Digit-Shifting Constants, Exponent, Exponent Laws, Exponential Function, Exponentiation, Faulhaber's Formula, Figurate Number, Moessner's Theorem, Narcissistic Number, Odd Power, Perfect Power, Power Rule, Power Tower, Square Number, Square Root, Squared, Sum, Truncated Power Function, Waring's Problem, Zero Explore this topic in the MathWorld classroom

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Barbeau, E. J. Power Play: A Country Walk through the Magical World of Numbers. Washington, DC: Math. Assoc. Amer., 1997.Beyer, W. H. "Laws of Exponents." CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, pp. 158 and 223, 1987.Derbyshire, J. Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics. New York: Penguin, 2004.Guy, R. K. "Diophantine Equations." Ch. D in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 137, 139-198, and 153-154, 1994.Knuth, D. E. "Two Notes on Notation." Amer. Math. Monthly 99, 403-422, 1992.Knuth, D. E. The Art of Computer Programming, Vol. 1: Fundamental Algorithms, 3rd ed. Reading, MA: Addison-Wesley, p. 57, 1997.Ribenboim, P. "Catalan's Conjecture." Amer. Math. Monthly 103, 529-538, 1996.Sloane, N. J. A. Sequences A001661/M5393, A005574/M1010, A052409, A052410, and A073084 in "The On-Line Encyclopedia of Integer Sequences."Spanier, J. and Oldham, K. B. "The Integer Powers (bx+c)^n and x^n" and "The Noninteger Powers x^nu." Ch. 11 and 13 in An Atlas of Functions. Washington, DC: Hemisphere, pp. 83-90 and 99-106, 1987.

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Weisstein, Eric W. "Power." From MathWorld--A Wolfram Web Resource.

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