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.
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.
In 2004, I was working on the bridge of a ship that was going to be inspected in a few days. There was a small hole in a wall about the size of an electrical outlet with exposed wiring that I wanted to hide from the inspectors. I printed out a picture of a kitten and taped it over the hole. I visited the ship again in 2018, and the kitten was still there.