Figuring out all the ways that 250/243 manifests itself

A few random ones that come to mind:

1) two 10/9's ≈ 6/5
2) three 10/9's ≈ 4/3
3) three 6/5's ≈ 16/9
4) 81/80 ≈ 25/24
5) 6/5 * 6/5 / 4/3 ≈ 10/9

I'm sure there are others too. All of these would seem, to a novice,
to be very foreign and unexpected things if you're coming off from
meantone, but they're all rooted in the same basic thing, which is
that 250/243 vanishes. Well, and some mysterious categorical thing
which we don't know, but you have to start somewhere.

It would be really helpful to make a table with all of these
relationships, but it's a bit hard to define what "all of these
relationships" really is. Does anyone see any ways to lay this sort of
thing out? I guess the goal is to find instances when a^m ≈ b^n, for a
and b which are useful to know about, and when m and n are small.
However, brute forcing from a huge list of 5-limit intervals would
take quite a while.

-Mike