In arithmetic and algebra the eighth power of a number n is the result of multiplying eight instances of n together. So:

n8 = n × n × n × n × n × n × n × n.

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:

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 ... (sequence A001016 in the OEIS)

In the archaic notation of Robert Recorde, the eighth power of a number was called the "zenzizenzizenzic".[1]

Algebra and number theory

Polynomial equations of degree 8 are octic equations. These have the form

${\displaystyle ax^{8}+bx^{7}+cx^{6}+dx^{5}+ex^{4}+fx^{3}+gx^{2}+hx+k=0.\,}$

The smallest known eighth power that can be written as a sum of eight eighth powers is[2]

${\displaystyle 1409^{8}=1324^{8}+1190^{8}+1088^{8}+748^{8}+524^{8}+478^{8}+223^{8}+90^{8}.}$

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:

${\displaystyle \zeta (8)={\frac {1}{1^{8))}+{\frac {1}{2^{8))}+{\frac {1}{3^{8))}+\cdots ={\frac {\pi ^{8)){9450))=1.00407\dots }$ ()

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:

${\displaystyle \zeta (2n)=(-1)^{n+1}{\frac {B_{2n}(2\pi )^{2n)){2(2n)!)).}$

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.[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]