jmc

There are $8\times 8=64$ combinations to examine. If one die is $1$, then even with an $8$ on the other die, no combinations will work. If one die is $2$, then even with an $8$ on the other die, no combinations will work. If one die is $3$, then even with an $8$ on the other die, no combinations will work. If one die is $4$, then even with an $8$ on the other die, no combinations will work. If one die is $5$, then the other die must be an $8$ to have a product over $36$. Thus, $(5,8)$ works. If one die is $6$, then the other die must be either $7$ or $8$ to have a product over $36$. Thus, $(6,7)$ and $(6,8)$ both work. If one die is $7$, then the other die can be $6,7,$ or $8$ to have a product over $36$. Thus, $(7,6)$, $(7,7)$, and $(7,8)$ all work. If one die is $8$, then the other die can be $5,6,7,$ or $8$ to have a product over $36$. Thus, $(8,5),(8,6),(8,7),$ and $(8,8)$ work. There are a total of $1+2+3+4=10$ combinations that work out of a total of $64$ possibilities. Thus, the answer is $\frac{10}{64}=\frac{5}{32}$, and the answer is $\boxed{\frac{5}{32}}$ But this may be slightly complicated, so you can also do solution 2 (scroll to bottom).