Using "// FullSimplify" with replace gives questionable answer

Questions
1

Hi;
When I replace {x,y,z} with {u,v,w} in the equation below, the answer does not seem to be exactly correct. For example, shouldn’t Sqrt[v^2] reduce to simply v and (w^4)^(1/4) reduce to w? Something that makes things ever more confusing is using the equals function “===” (Sqrt[v^2]===v) and (w^4)^(1/4)===w) both returns False.

x + Sqrt[y] + Sqrt[Sqrt[z]] == 1 /. {x -> u, y -> v^2, z -> w^4} // FullSimplify

u + Sqrt[v^2] + (w^4)^(1/4) == 1

Obviously, there is something here that I am not understanding so any clarification would be greatly appreciated.

Thanks,
Mitch Sandlin

SimplificationQuestion.nb (2.07 KB)

2

Of course they are False in the general case. If you assume the variables to be positive, then you should declare them so:

x + Sqrt[y] + Sqrt[Sqrt[z]] == 1 /. {x -> u, y -> v^2, z -> w^4}
Simplify[%, w > 0 && v > 0]

Another way to get what you expect is with PowerExpand:

x + Sqrt[y] + Sqrt[Sqrt[z]] == 1 /.
   {x -> u, y -> v^2, z -> w^4} // PowerExpand