Power
The exponent to which a given quantity is raised is known as its Power. The expression 
is therefore known as``
to the 
th Power.'' The rules for combining quantities containing powers arecalled the Exponent Laws.
Special names given to various powers are listed in the following table.
| Power | Name |
 | Square Root |
 | Cube Root |
| 2 | Squared |
| 3 | Cubed |
The Sum of 
th Powers of the first 
Positive Integersis given by Faulhaber's Formula,
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,... (Sloane's A001661).
Catalan's Conjecture 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. This Conjecture has not yet been proved orrefuted, although R. Tijdeman has proved that there can be only a finite number of exceptions should the Conjecturenot hold. It is also known that 8 and 9 are the only consecutive Cubic and Square Numbers (in either order). Hyyrö 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 , i.e., 
, 
, 
, .... The only Prime Numbers ofthe form 
for and Prime correspond to 
with 
, 2, 4, 6, 10, 14, 16, 20, 24,26, ... (Sloane's A005574).
There are no nontrivial solutions to the equation
for
(Guy 1994, p. 153).See also Apocalyptic Number, Biquadratic Number, Catalan's Conjecture, Catalan's Diophantine Problem,Cube Root, Cubed, Cubic Number, Exponent, Exponent Laws, Faulhaber's Formula,Figurate Number, Moessner's Theorem, Narcissistic Number, Power Rule, Square Number,Square Root, Squared, Sum, Waring's Problem
References
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, p. 158, 1987. 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. Ribenboim, P. ``Catalan's Conjecture.'' Amer. Math. Monthly 103, 529-538, 1996. Sloane, N. J. A. SequencesA001661/M5393and A005574/M1010in ``An On-Line Version of the Encyclopedia of Integer Sequences.''http://www.research.att.com/~njas/sequences/eisonline.html and Sloane, N. J. A. and Plouffe, S.The Encyclopedia of Integer Sequences. San Diego: Academic Press, 1995. Spanier, J. and Oldham, K. B. ``The Integer Powers 
and 
'' and ``The Noninteger Powers 
.'' Ch. 11 and 13 in An Atlas of Functions. Washington, DC: Hemisphere, pp. 83-90 and 99-106, 1987.
- Diabolical Cube
- Diabolical Square
- Diabolo
- Diacaustic
- Diagonalization
- Diagonal Matrix
- Diagonal Metric
- Diagonal (Polygon)
- Diagonal (Polyhedron)
- Diagonal Ramsey Number
- Diagonal Slash
- Diagonal (Solidus)
- Diagonals Problem
- Diagram
- Diameter
- Diameter (General)
- Diameter (Graph)
- Diamond
- Dice
- Dichroic Polynomial
- Dido's Problem
- Diesis
- Diffeomorphism
- Difference
- Difference Equation