It's hard to let things go. But if we're ever going to teach deeper and better, we've got teach fewer topics. So here's the beginning of a list:
Rationalizing the denominator. Time to kill it. Call it dead.
A bit of history: this topic only exists because armed with a square root table and a piece of paper, it is easier to scratch out root-2 divided by 2 than 1 divided by root-2. In other words, this topic has the same shelf life as interpolating on log tables. (Which I learned. Despite the TI SR-52 sitting on my desk.)
You say some nice things about this topic. You can connect it to other ideas. But it doesn't have a point and its time to kill it.