You can't multiply both sides by dx in much the same way you can't differentiate a duck. That said, even pure mathematicians sort of think of it that way as a useful shorthand.
I was trying to think of one. Lech Walesa came to mind, but it's doubtful. There was a Romanian president maybe 20 years ago that was a merchant ship officer, who often get f/l qualified (I did), but iirc, he flamed out.
It's been a while, but I think I remember this one. Lim 1/n =0 as n approaches infinity. Let x^0 be undefined. For any e>0 there exists an n such that |x^(1/n) -1| < e. If you desire x^(1/n) to be continuous at 0, you define x^0 as 1.
E2a: since x^(1/n)>1, you can drop the abs bars. I think you can get an inequality to pick n using logs.