The lack of handling root functions in hp prime

[parisse]

simplify(sqrt(x+y+2*sqrt(x*y))|x>0, y>0):
It looks simple, because *you* know how to do the simplification, but it is hard to have an algorithm that can solve that: you have to compute with algebraic extensions of Q[x,y]. The best I can do in Xcas (after fixing some code about assumptions) is return 0 for simplify(sqrt(x+y+2*sqrt(x*y))-sqrt(x)-sqrt(y)|x>0, y>0). It's slow on a desktop, therefore it won't work on the Prime.
In the second example, you probably meant
simplify(λ*√((xa)^2+y^2)+μ*√(x^2+(yb)^2)|yb=sqrt(r^2-x^2))
not y, and there the result seems OK.
solve expects polynomial-like equations. It's (again) very hard to handle sqrt or other fractional exponents.