Eighth power

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:

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

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

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

• Seventh power
• Sixth power
• Fifth power (algebra)
• Fourth power
• Cube (algebra)
• Square number

References

1. Womack, D. (2015), "Beyond tetration operations: their past, present and future", Mathematics in School, 44 (1): 23–26^{[dead link]}
2. Quoted in Meyrignac, Jean-Charles (2001-02-14). "Computing Minimal Equal Sums Of Like Powers: Best Known Solutions". Retrieved 2019-12-18.
3. Lighthill, M. J. (1952). "On sound generated aerodynamically. I. General theory". Proc. R. Soc. Lond. A. 211 (1107): 564–587. Bibcode:1952RSPSA.211..564L. doi:10.1098/rspa.1952.0060. S2CID 124316233.
4. Lighthill, M. J. (1954). "On sound generated aerodynamically. II. Turbulence as a source of sound". Proc. R. Soc. Lond. A. 222 (1148): 1–32. Bibcode:1954RSPSA.222....1L. doi:10.1098/rspa.1954.0049. S2CID 123268161.
5. Kardar, Mehran (2007). Statistical Physics of Fields. Cambridge University Press. p. 148. ISBN 978-0-521-87341-3. OCLC 1026157552.
6. Casimir, H. B. G.; Polder, D. (1948). "The influence of retardation on the London-van der Waals forces". Physical Review. 73 (4): 360. Bibcode:1948PhRv...73..360C. doi:10.1103/PhysRev.73.360.
7. Derjaguin, Boris V. (1960). "The force between molecules". Scientific American. 203 (1): 47–53. Bibcode:1960SciAm.203a..47D. doi:10.1038/scientificamerican0760-47. JSTOR 2490543.