$\left\{\begin{array}{l} \sqrt[3]{a}.b=2^5\\ a.\sqrt[3]{b}=2^7\end{array} \right.\\ \Leftrightarrow \left\{\begin{array}{l} a.b^3=2^15\\ a.\sqrt[3]{b}=2^7\end{array} \right.\\ \Leftrightarrow \left\{\begin{array}{l} a=\dfrac{2^15}{b^3}\\ \dfrac{a.b^3}{a.\sqrt[3]{b}}=\dfrac{2^15}{2^7}\end{array} \right.\\ \Leftrightarrow \left\{\begin{array}{l} a=\dfrac{2^15}{b^3}\\ \sqrt[3]{b^8}=2^8\end{array} \right.\\ \Leftrightarrow \left\{\begin{array}{l} a=\dfrac{2^15}{b^3}\\ \sqrt[3]{b}^8=2^8\end{array} \right.\\ \Leftrightarrow \left\{\begin{array}{l} a=\dfrac{2^15}{b^3}\\ \sqrt[3]{b}=2\end{array} \right.\\ \Leftrightarrow \left\{\begin{array}{l} a=\dfrac{2^15}{(2^3)^3}\\ b=2^3\end{array} \right.\\ \Leftrightarrow \left\{\begin{array}{l} a=64\\ b=8\end{array} \right.$