Express the following fraction in simplest form, only using positive exponents. \[ \frac{\left(-3 s^{-1} c^{-4}\right)^{2}}{4 s^{10} c^{-10}} \]

To simplify the expression \(\frac{\left(-3 s^{-1} c^{-4}\right)^{2}}{4 s^{10} c^{-10}}\), we start by simplifying the numerator: \[ \left(-3 s^{-1} c^{-4}\right)^{2} = (-3)^{2} (s^{-1})^{2} (c^{-4})^{2} = 9 s^{-2} c^{-8}. \] Now, substitute the numerator back into the fraction: \[ \frac{9 s^{-2} c^{-8}}{4 s^{10} c^{-10}}. \] Next, we can split the fraction into separate parts for the coefficients and the variables: \[ \frac{9}{4} \cdot \frac{s^{-2}}{s^{10}} \cdot \frac{c^{-8}}{c^{-10}}. \] We simplify the fraction with the coefficients: \[ \frac{9}{4} \] For the \(s\) terms, we use the property that \(\frac{s^{m}}{s^{n}} = s^{m-n}\): \[ \frac{s^{-2}}{s^{10}} = s^{-2-10} = s^{-12}. \] For the \(c\) terms: \[ \frac{c^{-8}}{c^{-10}} = c^{-8 - (-10)} = c^{2}. \] Now putting everything together, we have: \[ \frac{9}{4} s^{-12} c^{2}. \] To express the final answer using only positive exponents, we rewrite the \(s^{-12}\) term: \[ \frac{9 c^{2}}{4 s^{12}}. \] Thus, the expression in simplest form is \[ \boxed{\frac{9 c^{2}}{4 s^{12}}}. \]