Eighth power explained

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:

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, 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

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.}

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:

\zeta( 8 ) = (1)/(1 ^{8}) + (1)/(2 ^{8}) + (1)/(3 ^{8}) + … = (\pi ^{8})/(9450) = 1 . 00407 ... 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:\zeta( 2 n) = ( -1 ) ^{n+1} (B _{2n} ( 2 \pi) ^{2n})/(2 ( 2 n)!) .

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

Notes and References

1. Womack . David . 1 . Mathematics in School . 24767659 . 23–26 . Beyond tetration operations: their past, present and future . 44 . 2015.
2. Quoted in Web site: Meyrignac. Jean-Charles. Computing Minimal Equal Sums Of Like Powers: Best Known Solutions. 2001-02-14. 2019-12-18.
3. Lighthill . M. J. . James Lighthill . 1952 . On sound generated aerodynamically. I. General theory . . 211 . 1107 . 564–587. 10.1098/rspa.1952.0060 . 1952RSPSA.211..564L . 124316233 .
4. Lighthill . M. J. . James Lighthill . 1954 . On sound generated aerodynamically. II. Turbulence as a source of sound . . 222 . 1148 . 1–32. 10.1098/rspa.1954.0049 . 1954RSPSA.222....1L . 123268161 .
5. Book: Kardar, Mehran . Mehran Kardar . Statistical Physics of Fields . limited . 148 . 2007 . Cambridge University Press . 978-0-521-87341-3 . 1026157552.
6. Casimir . H. B. G. . Polder . D. . Hendrik Casimir . Dirk Polder . The influence of retardation on the London-van der Waals forces . . 73 . 4 . 1948 . 360 . 10.1103/PhysRev.73.360. 1948PhRv...73..360C .
7. Derjaguin . Boris V. . Boris Derjaguin . . The force between molecules . 203 . 1 . 1960 . 47–53 . 10.1038/scientificamerican0760-47 . 2490543. 1960SciAm.203a..47D .