There was a time I thought radicals should be simplified. A factor of a radicand should never be a perfect square. To do otherwise was just sloppy math—so I thought. Now, I think differently. You should consider it too.
Making the square root of 40 look like 2 on the square root of 10 serves no real purpose in the mathematics of real life. You can readily estimate the square root of 45, but trying to do that with three square root 5 is a much more complicated task, and for what? If I am going to the fabric store and I am asked how much ribbon I want, I better not say, “4 root 2” and expect to get the correct amount. At the hardware store, I am far better reasoning that the square root of 32 feet would be a bit less than 6 feet, and a bit more than 5 and one-half feet and just say 2 yards. The lumber department does not want to hear this nonsense about radicals and square roots. They want to cut my lumber and send me to the register to check out so they can help the next person also get a reasonable amount of lumber.
Now, I know, one needs to simplify radicals to combine radicals via addition, such as the square root of 27 plus the square root of 12, but seriously. This is not reality. This is a contrived problem that I have never seen come up in real life. Ever. And I sew and measure and do real life things with math—at home. It does not come up. It’s clever, like a party trick, but not terribly useful.
I do make certain my Math 2 students can “simplify radials,” but just for the “man.” Not for real life. I used to “ding” my students (take off 1 point just to be mildly irritating to get them to conform to convention) for not simplifying radicals. I am totally rethinking that.
Reality says leave radicals as they are so they are easy to estimate to be useful and to check for reasonableness. Done.