Answer
$${\left( {{1 \over {{y^5}}} + {y^5}} \right)^4} = {1 \over {{y^{20}}}} + {4 \over {{y^{10}}}} + 6 + 4{y^{10}}\, + {y^{20}}$$
Work Step by Step
$$\eqalign{
& {\left( {{1 \over {{y^5}}} + {y^5}} \right)^4} \cr
& {\rm{Apply\, the \,binomial \,theorem}} \cr
& {\left( {{1 \over {{y^5}}} + {y^5}} \right)^4} = {\left( {{1 \over {{y^5}}}} \right)^4} + \left( \matrix{
4 \hfill \cr
1 \hfill \cr} \right){\left( {{1 \over {{y^5}}}} \right)^3}\left( {{y^5}} \right) + \left( \matrix{
4 \hfill \cr
2 \hfill \cr} \right){\left( {{1 \over {{y^5}}}} \right)^2}{\left( {{y^5}} \right)^2} \cr
& \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, + \left( \matrix{
4 \hfill \cr
3 \hfill \cr} \right)\left( {{1 \over {{y^5}}}} \right){\left( {{y^5}} \right)^3}\, + {\left( {{y^5}} \right)^4} \cr
& {\rm{Evaluate\, each \,binomial\,coefficient\, use }}\left( \matrix{
n \hfill \cr
r \hfill \cr} \right) = {{n!} \over {\left( {n - r} \right)!r!}} \cr
& {\left( {{1 \over {{y^5}}} + {y^5}} \right)^4} = {\left( {{1 \over {{y^5}}}} \right)^4} + {{4!} \over {3!1!}}{\left( {{1 \over {{y^5}}}} \right)^3}\left( {{y^5}} \right) + {{4!} \over {2!2!}}{\left( {{1 \over {{y^5}}}} \right)^2}{\left( {{y^5}} \right)^2} \cr
& \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, + {{4!} \over {1!3!}}\left( {{1 \over {{y^5}}}} \right){\left( {{y^5}} \right)^3}\,\, + {\left( {{y^5}} \right)^4} \cr
& {\rm{Simplify}} \cr
& {\left( {{1 \over {{y^5}}} + {y^5}} \right)^4} = {\left( {{1 \over {{y^5}}}} \right)^4} + 4{\left( {{1 \over {{y^5}}}} \right)^3}\left( {{y^5}} \right) + 6{\left( {{1 \over {{y^5}}}} \right)^2}{\left( {{y^5}} \right)^2} \cr
& \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, + 4\left( {{1 \over {{y^5}}}} \right){\left( {{y^5}} \right)^3}\,\,\, + {\left( {{y^5}} \right)^4} \cr
& {\left( {{1 \over {{y^5}}} + {y^5}} \right)^4} = {1 \over {{y^{20}}}} + {4 \over {{y^{10}}}} + 6 + 4{y^{10}}\, + {y^{20}} \cr} $$
