In
arithmetic
Arithmetic () is an elementary part of mathematics that consists of the study of the properties of the traditional operations on numbers—addition, subtraction, multiplication, division, exponentiation, and extraction of roots. In the 19th c ...
and
algebra
Algebra () is one of the areas of mathematics, broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathem ...
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
In arithmetic and algebra the seventh power of a number ''n'' is the result of multiplying seven instances of ''n'' together. So:
:.
Seventh powers are also formed by multiplying a number by its sixth power, the square of a number by its fifth ...
, or the
fourth power
In arithmetic and algebra, the fourth power of a number ''n'' is the result of multiplying four instances of ''n'' together. So:
:''n''4 = ''n'' × ''n'' × ''n'' × ''n''
Fourth powers are also formed by multiplying a number by its cube. Furthe ...
of a number by itself.
The sequence of eighth powers of
integer
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...
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 an Anglo-Welsh physician and mathematician. He invented the equals sign (=) and also introduced the pre-existing plus sign (+) to English speakers in 1557.
Biography
Born around 1512, Robert Recorde was the second and last ...
, the eighth power of a number was called the "
zenzizenzizenzic".
Algebra and number theory
Polynomial
In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An ex ...
equations of
degree
Degree may refer to:
As a unit of measurement
* Degree (angle), a unit of angle measurement
** Degree of geographical latitude
** Degree of geographical longitude
* Degree symbol (°), a notation used in science, engineering, and mathemati ...
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 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 numbers:
:
Physics
In
aeroacoustics,
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 or Ising-Lenz model), named after the physicists Ernst Ising and Wilhelm Lenz, is a mathematical model of ferromagnetism in statistical mechanics. The model consists of discrete variables that represent ...
exhibits an inverse eighth power dependence of the
order parameter upon the
reduced temperature.
The
Casimir–Polder force between two molecules decays as the inverse eighth power of the distance between them.
See also
*
Seventh power
In arithmetic and algebra the seventh power of a number ''n'' is the result of multiplying seven instances of ''n'' together. So:
:.
Seventh powers are also formed by multiplying a number by its sixth power, the square of a number by its fifth ...
*
Sixth power In arithmetic and algebra the sixth power of a number ''n'' is the result of multiplying six instances of ''n'' together. So:
:.
Sixth powers can be formed by multiplying a number by its fifth power, multiplying the square of a number by its four ...
*
Fifth power (algebra)
In arithmetic and algebra, the fifth power or sursolid of a number ''n'' is the result of multiplying five instances of ''n'' together:
:.
Fifth powers are also formed by multiplying a number by its fourth power, or the square of a number by its ...
*
Fourth power
In arithmetic and algebra, the fourth power of a number ''n'' is the result of multiplying four instances of ''n'' together. So:
:''n''4 = ''n'' × ''n'' × ''n'' × ''n''
Fourth powers are also formed by multiplying a number by its cube. Furthe ...
*
Cube (algebra)
In arithmetic and algebra, the cube of a number is its third power, that is, the result of multiplying three instances of together.
The cube of a number or any other mathematical expression is denoted by a superscript 3, for example or ...
*
Square number
In mathematics, a square number or perfect square is an integer that is the square of an integer; in other words, it is the product of some integer with itself. For example, 9 is a square number, since it equals and can be written as .
The u ...
References
Integers
Number theory
Elementary arithmetic
Integer sequences
Unary operations
Figurate numbers
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