#### Answer

$5(x^4+z^{8})(x^2+z^{4})(x+z^{2})(x-z^{2})$

#### Work Step by Step

Factoring the $GCF=5,$ the given expression is equivalent to
\begin{array}{l}\require{cancel}
5x^8-5z^{16}
\\\\=
5(x^8-z^{16})
.\end{array}
The expressions $
x^8
$ and $
z^{16}
$ are both perfect squares (the square root is exact) and are separated by a minus sign. Hence, $
x^8-z^{16}
,$ is a difference of $2$ squares. Using the factoring of the difference of $2$ squares which is given by $a^2-b^2=(a+b)(a-b),$ the expression above is equivalent to
\begin{array}{l}\require{cancel}
5(x^8-z^{16})
\\\\=
5[(x^4)^2-(z^{8})^2]
\\\\=
5[(x^4+z^{8})(x^4-z^{8})]
\\\\=
5(x^4+z^{8})(x^4-z^{8})
\\\\=
5(x^4+z^{8})[(x^2)^2-(z^{4})^2]
\\\\=
5(x^4+z^{8})(x^2+z^{4})(x^2-z^{4})
\\\\=
5(x^4+z^{8})(x^2+z^{4})[(x)^2-(z^{2})^2]
\\\\=
5(x^4+z^{8})(x^2+z^{4})(x+z^{2})(x-z^{2})
.\end{array}