Simplify the following expression completely. Write your answer using only positive exponents. \[ \left(\frac{a^{-5} b^{4}}{a^{3} b^{-1}}\right)^{-5}=\square \]

To simplify the expression \(\left(\frac{a^{-5} b^{4}}{a^{3} b^{-1}}\right)^{-5}\), we start with the inner fraction. Recall that when dividing like bases, we subtract the exponents. So, for \(a\): \[ \frac{a^{-5}}{a^{3}} = a^{-5 - 3} = a^{-8} \] And for \(b\): \[ \frac{b^{4}}{b^{-1}} = b^{4 - (-1)} = b^{4 + 1} = b^{5} \] Thus, we can rewrite the inner expression: \[ \frac{a^{-5} b^{4}}{a^{3} b^{-1}} = a^{-8} b^{5} \] Now we will take this result to the power of \(-5\): \[ (a^{-8} b^{5})^{-5} = a^{-8 \cdot -5} b^{5 \cdot -5} = a^{40} b^{-25} \] Now, we write everything using only positive exponents. Recall that \(b^{-25} = \frac{1}{b^{25}}\): \[ a^{40} b^{-25} = \frac{a^{40}}{b^{25}} \] Therefore, the fully simplified expression is: \[ \frac{a^{40}}{b^{25}} \]