In arithmetic and algebra 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 integers is:
In the archaic notation of Robert Recorde, the eighth power of a number was called the "zenzizenzizenzic".[1]
Polynomial 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[2]
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:
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.[3][4]
The ordered phase of the two-dimensional Ising model exhibits an inverse eighth power dependence of the order parameter upon the reduced temperature.[5]
The Casimir–Polder force between two molecules decays as the inverse eighth power of the distance between them.[6][7]