By the way, on typical math tests and in typical math books, they call this stuff "simplifying" or "reducing" radicals...  But, the value doesn't get smaller, so I don't like the "reducing" word...

And, I ask you...  Which is "simpler?"

Which would take less time to punch into a calculator?

If I had my way (which I rarely do), I'd bad this process altogether.  But, it's in all the books and it will be on most algebra tests you'll run into.  So, let's just think of it as a game we need to play.