Fourth power

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

n⁴ = 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 biquadratic numbers or tesseractic numbers) is:

1, 16, 81, 256, 625, 1296, 2401, 4096, 6561, 10000, ... (sequence A000583 in OEIS)

The last two digits of a fourth power of an integer can be easily shown (for instance, by computing the squares of possible last two digits of square numbers) to be restricted to only twelve possibilities:

00, 01, 16, 21, 25, 36, 41, 56, 61, 76, 81, 96

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).

Euler conjectured a fourth power cannot be written as the sum of 3 smaller fourth powers, but 200 years later this was disproven with:

958004 + 2175194 + 4145604 = 4224814.
Equations containing a fourth power

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

Square (algebra)
Cube (algebra)
Exponentiation
Fifth power (algebra)
Perfect power

References

Weisstein, Eric W., "Biquadratic Number", MathWorld.

Mathematics Encyclopedia