Fourth Power

Get Fourth Power essential facts below. View Videos or join the Fourth Power discussion. Add Fourth Power to your PopFlock.com topic list for future reference or share this resource on social media.
## Properties

## Equations containing a fourth power

## See also

## References

This article uses material from the Wikipedia page available here. It is released under the Creative Commons Attribution-Share-Alike License 3.0.

Fourth Power

In arithmetic and algebra, the **fourth power** of a number *n* is the result of multiplying four instances of *n* together. So:

*n*^{4}=*n*×*n*×*n*×*n*

Fourth powers are also formed by multiplying a number by its cube. Furthermore, they are squares of squares.

The sequence of fourth powers of integers (also known as **biquadrates** or **tesseractic numbers**) is:

- 0, 1, 16, 81, 256, 625, 1296, 2401, 4096, 6561, 10000, 14641, 20736, 28561, 38416, 50625, 65536, 83521, 104976, 130321, 160000, 194481, 234256, 279841, 331776, 390625, 456976, 531441, 614656, 707281, 810000, ... (sequence in the OEIS)

The last two digits of a fourth power of an integer in senary or decimal can be easily shown (for instance, by computing the squares of possible last two digits of square numbers) to be restricted to only *eight* possibilities in senary, and only *twelve* possibilities in decimal.

- In senary

- if a number ends in 0, its fourth power ends in (in fact in )
- if a number ends in 1 or 5, its fourth power ends in , or
- if a number ends in 2 or 4, its fourth power ends in , or
- if a number ends in 3, its fourth power ends in (in fact in )

- In decimal

- if a number ends in 0, its fourth power ends in (in fact in )
- if a number ends in 1, 3, 7 or 9 its fourth power ends in , , , or
- if a number ends in 2, 4, 6, or 8 its fourth power ends in , , , or
- if a number ends in 5 its fourth power ends in (in fact in )

- These twelve possibilities can be conveniently expressed as 00,
*e*1,*o*6 or 25 where*o*is an odd digit and*e*an even digit.

Every positive integer can be expressed as the sum of at most 19 fourth powers; every sufficiently large integer can be expressed as the sum of at most 16 fourth powers (see Waring's problem).

Fermat knew that a fourth power cannot be the sum of two other fourth powers (the *n*=4 case of Fermat's Last Theorem; see Fermat's right triangle theorem). Euler conjectured that a fourth power cannot be written as the sum of three fourth powers, but 200 years later, in 1986, this was disproven by Elkies with:

Elkies showed that there are infinitely many other counterexamples for exponent four, some of which are:^{[1]}

- (Allan MacLeod)
- (D.J. Bernstein)
- (D.J. Bernstein)
- (D.J. Bernstein)
- (D.J. Bernstein)
- (Roger Frye, 1988)
- (Allan MacLeod,1998)

That the equation *x*^{4} + *y*^{4} = *z*^{4} has no solutions in nonzero integers (a special case of Fermat's Last Theorem), was known by Fermat; see Fermat's right triangle theorem.

Fourth-degree equations, which contain a fourth degree (but no higher) polynomial are, by the Abel-Ruffini theorem, the highest degree equations having a general solution using radicals.

- Square (algebra)
- Cube (algebra)
- Exponentiation
- Fifth power (algebra)
- Sixth power
- Seventh power
- Perfect power

**^**Quoted in Meyrignac, Jean-Charles (14 February 2001). "Computing Minimal Equal Sums Of Like Powers: Best Known Solutions". Retrieved 2017.

This article uses material from the Wikipedia page available here. It is released under the Creative Commons Attribution-Share-Alike License 3.0.

Popular Products

Music Scenes

Popular Artists