Answer
$0$
Work Step by Step
For any matrix \[
\left[ {\begin{array}{cc}
a & b \\
c & d \\
\end{array} } \right]
\] the determinant equals $ad-bc$. For the matrix \[
\left[ {\begin{array}{cc}
e^{-3} & 3e^{10} \\
2e^{-5} & 6e^8 \\
\end{array} } \right]
\] the determinant equals $e^{-3}(6e^{8})-(2e^{-5})(3e^{10})=6e^5-6e^5=0.$