In
arithmetic
Arithmetic is an elementary branch of mathematics that deals with numerical operations like addition, subtraction, multiplication, and division. In a wider sense, it also includes exponentiation, extraction of roots, and taking logarithms.
...
and
algebra
Algebra is a branch of mathematics that deals with abstract systems, known as algebraic structures, and the manipulation of expressions within those systems. It is a generalization of arithmetic that introduces variables and algebraic ope ...
, the eighth power of a number ''n'' is the result of multiplying eight instances of ''n'' together. So:
:.
Eighth
powers are also formed by multiplying a number by its
seventh power, or the
fourth power of a number by itself.
The sequence of eighth powers of
integer
An integer is the number zero (0), a positive natural number (1, 2, 3, ...), or the negation of a positive natural number (−1, −2, −3, ...). The negations or additive inverses of the positive natural numbers are referred to as negative in ...
s is:
:0, 1, 256, 6561, 65536, 390625, 1679616, 5764801, 16777216, 43046721, 100000000, 214358881, 429981696, 815730721, 1475789056, 2562890625, 4294967296, 6975757441, 11019960576, 16983563041, 25600000000, 37822859361, 54875873536, 78310985281, 110075314176, 152587890625 ...
In the
archaic notation of
Robert Recorde
Robert Recorde () was a Welsh physician and mathematician. He invented the equals sign (=) and also introduced the pre-existing plus (+) and minus (−) signs to English speakers in 1557.
Biography
Born around 1510, Robert Recorde was the sec ...
, the eighth power of a number was called the "
zenzizenzizenzic Zenzizenzizenzic is an obsolete form of mathematical notation representing the eighth power of a number (that is, the zenzizenzizenzic of ''x'' is ''x''8), dating from a time when powers were written out in words rather than as superscript numbers. ...
".
Algebra and number theory
Polynomial
In mathematics, a polynomial is a Expression (mathematics), mathematical expression consisting of indeterminate (variable), indeterminates (also called variable (mathematics), variables) and coefficients, that involves only the operations of addit ...
equations of
degree 8 are
octic equations. These have the form
:
The smallest known eighth power that can be written as a sum of eight eighth powers is
[Quoted in ]
:
The sum of the reciprocals of the nonzero eighth powers is the
Riemann zeta function
The Riemann zeta function or Euler–Riemann zeta function, denoted by the Greek letter (zeta), is a mathematical function of a complex variable defined as \zeta(s) = \sum_^\infty \frac = \frac + \frac + \frac + \cdots for and its analytic c ...
evaluated at 8, which can be expressed in terms of the eighth power of
pi:
:
()
This is an example of a more general expression for
evaluating the Riemann zeta function at positive even integers, in terms of the
Bernoulli number
In mathematics, the Bernoulli numbers are a sequence of rational numbers which occur frequently in analysis. The Bernoulli numbers appear in (and can be defined by) the Taylor series expansions of the tangent and hyperbolic tangent function ...
s:
:
Physics
In
aeroacoustics Aeroacoustics is a branch of acoustics that studies noise generation via either turbulent fluid motion or aerodynamic forces interacting with surfaces. Noise generation can also be associated with periodically varying flows. A notable example of t ...
,
Lighthill's eighth power law states that the power of the sound created by a turbulent motion, far from the turbulence, is proportional to the eighth power of the characteristic turbulent velocity.
The ordered phase of the two-dimensional
Ising model
The Ising model (or Lenz–Ising model), named after the physicists Ernst Ising and Wilhelm Lenz, is a mathematical models in physics, mathematical model of ferromagnetism in statistical mechanics. The model consists of discrete variables that r ...
exhibits an inverse eighth power dependence of the
order parameter
In physics, chemistry, and other related fields like biology, a phase transition (or phase change) is the physical process of transition between one state of a medium and another. Commonly the term is used to refer to changes among the basic s ...
upon the
reduced temperature
In thermodynamics, the reduced properties of a fluid are a set of state variables scaled by the fluid's state properties at its critical point. These dimensionless thermodynamic coordinates, taken together with a substance's compressibility fact ...
.
The
Casimir–Polder force between two molecules decays as the inverse eighth power of the distance between them.
See also
*
Seventh power
*
Sixth power
*
Fifth power
*
Fourth power
*
Cube
A cube or regular hexahedron is a three-dimensional space, three-dimensional solid object in geometry, which is bounded by six congruent square (geometry), square faces, a type of polyhedron. It has twelve congruent edges and eight vertices. It i ...
*
Square number
In mathematics, a square number or perfect square is an integer that is the square (algebra), square of an integer; in other words, it is the multiplication, product of some integer with itself. For example, 9 is a square number, since it equals ...
References
{{Classes of natural numbers
Integers
Number theory
Elementary arithmetic
Integer sequences
Unary operations
Figurate numbers