Answer
$${\left( {{1 \over {{x^4}}} + {x^4}} \right)^4} = {1 \over {{x^{16}}}} + {4 \over {{x^8}}} + 6 + 4{x^8}\, + {x^{16}}$$
Work Step by Step
$$\eqalign{
& {\left( {{1 \over {{x^4}}} + {x^4}} \right)^4} \cr
& {\rm{Apply\, the binomial \,theorem}} \cr
& {\left( {{1 \over {{x^4}}} + {x^4}} \right)^4} = {\left( {{1 \over {{x^4}}}} \right)^4} + \left( \matrix{
4 \hfill \cr
1 \hfill \cr} \right){\left( {{1 \over {{x^4}}}} \right)^3}\left( {{x^4}} \right) + \left( \matrix{
4 \hfill \cr
2 \hfill \cr} \right){\left( {{1 \over {{x^4}}}} \right)^2}{\left( {{x^4}} \right)^2} \cr
& \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, + \left( \matrix{
4 \hfill \cr
3 \hfill \cr} \right)\left( {{1 \over {{x^4}}}} \right){\left( {{x^4}} \right)^3}\,\, + {\left( {{x^4}} \right)^4} \cr
& {\rm{Evaluate\, each\, binomial\,coefficient \,use }}\left( \matrix{
n \hfill \cr
r \hfill \cr} \right) = {{n!} \over {\left( {n - r} \right)!r!}} \cr
& {\left( {{1 \over {{x^4}}} + {x^4}} \right)^4} = {\left( {{1 \over {{x^4}}}} \right)^4} + {{4!} \over {3!1!}}{\left( {{1 \over {{x^4}}}} \right)^3}\left( {{x^4}} \right) + {{4!} \over {2!2!}}{\left( {{1 \over {{x^4}}}} \right)^2}{\left( {{x^4}} \right)^2} \cr
& \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, + {{4!} \over {1!3!}}\left( {{1 \over {{x^4}}}} \right){\left( {{x^4}} \right)^3}\, + {\left( {{x^4}} \right)^4} \cr
& {\rm{Simplify}} \cr
& {\left( {{1 \over {{x^4}}} + {x^4}} \right)^4} = {\left( {{1 \over {{x^4}}}} \right)^4} + 4{\left( {{1 \over {{x^4}}}} \right)^3}\left( {{x^4}} \right) + 6{\left( {{1 \over {{x^4}}}} \right)^2}{\left( {{x^4}} \right)^2} + 4\left( {{1 \over {{x^4}}}} \right){\left( {{x^4}} \right)^3}\, \cr
& \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, + {\left( {{x^4}} \right)^4} \cr
& {\left( {{1 \over {{x^4}}} + {x^4}} \right)^4} = {1 \over {{x^{16}}}} + {4 \over {{x^8}}} + 6 + 4{x^8}\, + {x^{16}} \cr} $$
