Inarithmetic and algebra, the fourth power of a number n is the result of multiplying four instances of n together. So:
Fourth powers are also formed by multiplying a number by its cube. Furthermore, they are squares of squares.
Some people refer to n4 as n “tesseracted”, “hypercubed”, “zenzizenzic”, “biquadrate” or “supercubed” instead of “to the power of 4”.
The sequence of fourth powers of integers (also known as biquadratesortesseractic numbers) is:
The last digit of a fourth power in decimal can only be 0 (in fact 0000), 1, 5 (in fact 0625), or 6.
Every positive integer can be expressed as the sum of at most 19 fourth powers; every integer larger than 13792 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 caseofFermat'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]
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.