jmc

Note that $144$ is a multiple of $2^4$ and $3^2$ since $$144=16\cdot 9=2^4\cdot 3^2.$$Note that $2^5=32$ is not a factor of $144$ since dividing $144$ by $32$ gives a remainder of $16$. Similarly, $3^3=27$ is not a factor of $144$ since dividing $144$ by $27$ gives a remainder of $9$.

It follows that $2^4$ is the greatest power of $2$ that is a factor of $144$, and that $3^2$ is the greatest power of $3$ that is a factor of $144$. So $x=4$ and $y=2$. So our final answer is $$\begin{array}{rl}{\left(\frac{1}{5}\right)}^{2-4}& ={\left(\frac{1}{5}\right)}^{-2}\\ & ={\left({\left(\frac{1}{5}\right)}^{-1}\right)}^2\\ & =5^2\\ & =\boxed{25}.\end{array}$$