a. \( \sqrt{625 n^{4} \sqrt[3]{4 m^{2}}} \)

To simplify \( \sqrt{625 n^{4} \sqrt[3]{4 m^{2}}} \), we start with the square root of \( 625 \), which is \( 25 \). Next, for \( n^{4} \), we find the square root is \( n^{2} \). As for the term \( \sqrt[3]{4 m^{2}} \), we keep it in its current form for the moment. Therefore, our expression becomes: \[ \sqrt{625 n^{4} \sqrt[3]{4 m^{2}}} = 25 n^{2} \sqrt{\sqrt[3]{4 m^{2}}} \] Now, to deal with \( \sqrt{\sqrt[3]{4 m^{2}}} \), we first rewrite it as \( (4 m^{2})^{1/6} \). Thus, the final simplified version of the original expression is: \[ 25 n^{2} (4 m^{2})^{1/6} \] And, if you want to expand that further, \( (4 m^{2})^{1/6} = 2^{1/3} m^{1/3} \), leading to: \[ 25 n^{2} 2^{1/3} m^{1/3} \] Voilà, you have your simplified expression!