What is complex number z if $z^8+16z^4+256=0$? So far, I have set y to equal to $z^4$ and used the quadratic equation to solve $y = -8+8\sqrt{3}i$ or $-8-8\sqrt{3}i$. How do I determine the 8 different values of $z$?

When you factorize the polynomial you'll get the following products:
$$(x^2 - 2 x + 4) (x^2 + 2 x + 4) (x^4 - 4 x^2 + 16) = 0 $$
And therefore you get the following roots by computing quadratic and quartic equations:
$-1-i\sqrt{3}$
$-1+i\sqrt{3}$
$1-i\sqrt{3}$
$1+i\sqrt{3}$
$-\sqrt{2-2i\sqrt{3}}$
$\sqrt{2-2i\sqrt{3}}$
$-\sqrt{2+2i\sqrt{3}}$
$\sqrt{2+2i\sqrt{3}}$
