Express the following fraction in simplest form, only using positive exponents. \[ \frac{20 y^{2} h^{-1}}{\left(5 y^{5} h^{2}\right)^{2}} \]

To simplify the fraction \(\frac{20 y^{2} h^{-1}}{\left(5 y^{5} h^{2}\right)^{2}}\), we start with the denominator. First, we calculate \(\left(5 y^{5} h^{2}\right)^{2}\): \[ \left(5 y^{5} h^{2}\right)^{2} = 5^{2} (y^{5})^{2} (h^{2})^{2} = 25 y^{10} h^{4}. \] Now, we can rewrite the original fraction: \[ \frac{20 y^{2} h^{-1}}{25 y^{10} h^{4}}. \] Next, we simplify the coefficients (numerators and denominators): \[ \frac{20}{25} = \frac{4}{5}. \] Now we simplify the \(y\) terms: \[ \frac{y^{2}}{y^{10}} = y^{2 - 10} = y^{-8}. \] And for the \(h\) terms: \[ \frac{h^{-1}}{h^{4}} = h^{-1 - 4} = h^{-5}. \] Putting everything together, we have: \[ \frac{4}{5} y^{-8} h^{-5}. \] Now, we express this with positive exponents: \[ \frac{4}{5} \frac{1}{y^{8} h^{5}} = \frac{4}{5 y^{8} h^{5}}. \] Thus, the simplified form of the fraction is: \[ \frac{4}{5 y^{8} h^{5}}. \]