The first mathematician: Frodo

The second mathematician: Saruman

Frodo has $x$ and Saruman has $y$. 

Frodo thinks that Saruman has either $6^{12/x}$ or $6^{12}/x$. 

Saruman thinks that Frodo has $6^{12/y}$ or $6^{12}/y$. 

I derived an answer based on that but I'm not sure if it's correct. The answer is:

>! The first one has $6^6$ and
>! the second one has $6^6$