Simplify the following and express your solution with no negative exponents. Include a full solution.
\[
\left(x^{3} y\right)^{-2}\left(\frac{x y^{3}}{x^{2} y^{2}}\right)^{2}
\]

Step 1 ：Simplify the expression inside the parentheses. For the first part, \((x^{3} y)^{-2}\), we can apply the rule \(a^{-n} = \frac{1}{a^n}\) to get \(\frac{1}{(x^{3} y)^{2}}\). Then, we can apply the rule \((a b)^n = a^n b^n\) to get \(\frac{1}{x^{6} y^{2}}\).

Step 2 ：For the second part, \(\left(\frac{x y^{3}}{x^{2} y^{2}}\right)^{2}\), we can simplify the fraction inside the parentheses first. \(x y^{3}\) divided by \(x^{2} y^{2}\) simplifies to \(\frac{y}{x}\). Then, we square this to get \(\left(\frac{y}{x}\right)^{2}\), which simplifies to \(\frac{y^{2}}{x^{2}}\).

Step 3 ：Finally, we multiply the two parts together. \(\frac{1}{x^{6} y^{2}}\) times \(\frac{y^{2}}{x^{2}}\) simplifies to \(\frac{1}{x^{4}}\).

Step 4 ：The simplified form of the given expression is \(x^{-8}\). However, the question asks for the solution with no negative exponents. We can convert \(x^{-8}\) to \(\frac{1}{x^{8}}\) to meet this requirement.

Step 5 ：Final Answer: \(\boxed{\frac{1}{x^{8}}}\)