In arithmetic and algebra the eighth power of a number n is the result of multiplying eight instances of n together. So:
n 8 = 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
a
x
8
+
b
x
7
+
c
x
6
+
d
x
5
+
e
x
4
+
f
x
3
+
g
x
2
+
h
x
+
k
=
0.
{\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]
1409
8
=
1324
8
+
1190
8
+
1088
8
+
748
8
+
524
8
+
478
8
+
223
8
+
90
8
.
{\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 :
ζ ζ -->
(
8
)
=
1
1
8
+
1
2
8
+
1
3
8
+
⋯ ⋯ -->
=
π π -->
8
9450
=
1.00407
… … -->
{\displaystyle \zeta (8)={\frac {1}{1^{8}}}+{\frac {1}{2^{8}}}+{\frac {1}{3^{8}}}+\cdots ={\frac {\pi ^{8}}{9450}}=1.00407\dots }
(OEIS : A013666 )
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 :
ζ ζ -->
(
2
n
)
=
(
− − -->
1
)
n
+
1
B
2
n
(
2
π π -->
)
2
n
2
(
2
n
)
!
.
{\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]
See also
References
Possessing a specific set of other numbers
Expressible via specific sums