Answer
$$243{x^2}\sqrt x - 405x\sqrt x + 270\sqrt x - {{90} \over {\sqrt x }} + {{15} \over {x\sqrt x }} - {1 \over {{x^2}\sqrt x }}$$
Work Step by Step
$$\eqalign{
& {\left( {3\sqrt x - {1 \over {\sqrt x }}} \right)^5} \cr
& {\left( {3\sqrt x - {1 \over {\sqrt x }}} \right)^5} = {\left( {3\sqrt x + \left( { - {1 \over {\sqrt x }}} \right)} \right)^5} \cr
& {\rm{Apply\, the\, binomial\, theorem}} \cr
& \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = {\left( {3\sqrt x } \right)^5} + \left( \matrix{
5 \hfill \cr
1 \hfill \cr} \right){\left( {3\sqrt x } \right)^4}\left( { - {1 \over {\sqrt x }}} \right) + \left( \matrix{
5 \hfill \cr
2 \hfill \cr} \right){\left( {3\sqrt x } \right)^3}{\left( { - {1 \over {\sqrt x }}} \right)^2} \cr
& \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, + \left( \matrix{
5 \hfill \cr
3 \hfill \cr} \right){\left( {3\sqrt x } \right)^2}{\left( { - {1 \over {\sqrt x }}} \right)^3}\, + \left( \matrix{
5 \hfill \cr
4 \hfill \cr} \right)\left( {3\sqrt x } \right){\left( { - {1 \over {\sqrt x }}} \right)^4}\,\, + \,{\left( { - {1 \over {\sqrt x }}} \right)^5} \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( {3\sqrt x } \right)^5} + {{5!} \over {4!1!}}{\left( {3\sqrt x } \right)^4}\left( { - {1 \over {\sqrt x }}} \right) + {{5!} \over {3!2!}}{\left( {3\sqrt x } \right)^3}{\left( { - {1 \over {\sqrt x }}} \right)^2} \cr
& \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, + {{5!} \over {2!3!}}{\left( {3\sqrt x } \right)^2}{\left( { - {1 \over {\sqrt x }}} \right)^3}\, + {{5!} \over {1!4!}}\left( {3\sqrt x } \right){\left( { - {1 \over {\sqrt x }}} \right)^4}\,\, + \,{\left( { - {1 \over {\sqrt x }}} \right)^5} \cr
& {\rm{Simplify}} \cr
& {\left( {3\sqrt x - {1 \over {\sqrt x }}} \right)^5} = {\left( {3\sqrt x } \right)^5} + 5{\left( {3\sqrt x } \right)^4}\left( { - {1 \over {\sqrt x }}} \right) + 10{\left( {3\sqrt x } \right)^3}{\left( { - {1 \over {\sqrt x }}} \right)^2} \cr
& \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, + 10{\left( {3\sqrt x } \right)^2}{\left( { - {1 \over {\sqrt x }}} \right)^3}\, + 5\left( {3\sqrt x } \right){\left( {{1 \over {\sqrt x }}} \right)^4}\,\, + \,{\left( { - {1 \over {\sqrt x }}} \right)^5} \cr
& {\left( {3\sqrt x - {1 \over {\sqrt x }}} \right)^5} = 243{x^2}\sqrt x - 405x\sqrt x + 270\sqrt x - {{90} \over {\sqrt x }} + {{15} \over {x\sqrt x }} - {1 \over {{x^2}\sqrt x }} \cr} $$
