Question 16514
While you didn't say what you wanted done with this expression...in fact you didn't say anything at all, but being the mind readers that we are, I would say that you want to factor this expression! It's the difference of two squares:
{{{(x^8)^2 - 1^2)}}} and can be factored as follows:

{{{x^16 - 1 =(x^8 - 1)(x^8 + 1)}}} But (x^8 - 1) is also the difference of two squares and it too can be factored.

{{{x^16 - 1 = (x^4 - 1)(x^4 + 1)(x^8 + 1)}}} Once again, the (x^4 - 1) is the difference of two squares and it can be factored as the others could.

{{{x^16 - 1 = (x^2 - 1)(x ^2 + 1)(x^4 + 1)(x^8 + 1)}}} and finally, the (x^2 - 1) can be factored.

{{{x^16 - 1 = (x- 1)(x + 1)(x^2 + 1)(x^4 + 1)(x^8 + 1)}}} The end!

We are done, except, that is, if you want to get into complex numbers, and I assume that you'd rather not.