Simplify. \[ \left(-4 b^{-2} c^{5}\right)^{2} \] Write your answer using only positive exponents. \( \square \)

When simplifying \( \left(-4 b^{-2} c^{5}\right)^{2} \), we need to apply the power to each component inside the parentheses: 1. First, square the coefficient: \(-4\) squared is \(16\). 2. Next, apply the power to \(b^{-2}\): \( (b^{-2})^{2} = b^{-4} \). 3. Finally, apply the power to \(c^{5}\): \( (c^{5})^{2} = c^{10} \). Putting it all together, we have: \[ 16 b^{-4} c^{10} \] To express it using positive exponents, we rewrite \(b^{-4}\) as \(\frac{1}{b^{4}}\): \[ \frac{16 c^{10}}{b^{4}} \] So the simplified form is: \[ \frac{16 c^{10}}{b^{4}} \]