The roots of x
^{2}  6x + 1 = 0 are [3+2*SQRT(2), 32*SQRT(2)].
The two expressions for the roots may be given as
[j+k] and [jk]
where k is the part of the expression that has the radical.
In the binomial expansion of [j+k]
^{n} where n is an even integer, the sign of the operation remains plus (+), and in the binomial expansion of [jk]
^{n} the sign of the operation alternates between minus () where k is raised to an odd power and plus (+) where it is raised to an even power. Thus, when the two exponentiated roots [j+k]
^{n} and [jk]
^{n} are added, the terms for the odd powers of k (the part of the expression with the radical) cancel each other out. As what remains are the even powers for k, the radical "disappears", leaving a rational number.
The final digit of A
^{n}+ B
^{n } has a cycle length of 6:
{6, 4, 8, 6, 4, 2}. As a number needs to end in either 0 or 5 to be divisible by 5, and as no 0 or 5 is in the cycle for A
^{n}+ B
^{n },
A
^{n}+ B
^{n }can not be divisible by 5.
Edited on June 30, 2011, 4:55 pm

Posted by Dej Mar
on 20110630 16:51:19 