Express the following fraction in simplest form, only using positive exponents. \[ \frac{\left(2 v^{-4} t^{-1}\right)^{-5}}{4 v^{-1} t} \]

To simplify the given fraction \(\frac{\left(2 v^{-4} t^{-1}\right)^{-5}}{4 v^{-1} t}\), we’ll start with the numerator. 1. First, simplify the numerator: \[ \left(2 v^{-4} t^{-1}\right)^{-5} = 2^{-5} (v^{-4})^{-5} (t^{-1})^{-5} = \frac{1}{2^5} v^{20} t^{5} = \frac{v^{20} t^{5}}{32}. \] 2. Now, we substitute this back into the fraction: \[ \frac{\frac{v^{20} t^{5}}{32}}{4 v^{-1} t}. \] 3. We can rewrite the denominator: \[ 4 v^{-1} t = 4 \cdot \frac{t}{v} = \frac{4t}{v}. \] 4. Now we can rewrite the fraction as: \[ \frac{v^{20} t^{5}}{32} \cdot \frac{v}{4t} = \frac{v^{20} t^{5} \cdot v}{32 \cdot 4t} = \frac{v^{21} t^{5}}{128t}. \] 5. Now simplify the fraction by dividing \(t^{5}\) by \(t\): \[ \frac{v^{21} t^{5}}{128t} = \frac{v^{21} t^{4}}{128}. \] Thus, the fraction in simplest form with only positive exponents is: \[ \frac{v^{21} t^{4}}{128}. \]