Solving some problem parametrically, I got the following answer:


$$ \dfrac{5x}{4} + \sqrt{\dfrac{y^2}{4} - \dfrac{x^2}{16}} + \dfrac{1}{10} \sqrt{10x^2 + 9y^2} + \dfrac{1}{5} \sqrt{5x^2 + 16y^2} $$


And I had a question: can there be a rational answer for some $ x, y \in \mathbb {R\_+} $? Well, or what “maximally” simple answer can you get where each term will be positive? I suspect that there will still be one root always for any values of $ x, y $. By a simple answer, I understand the number of radicals used after all the simplifications, for some reason you can somehow call it scientifically, but I don’t know how.



The existence of rational points
Take $x=1$ and $y^2=z$. Then conditions are that $z/4-1/16$, $10+9z$ and $5+16z$ are squares. By multiplying these conditions we get the elliptic curve $u^2 = (z/4-1/16)(10+9z)(5+16z)$. It has torsion group $Z/2Z \times Z/4Z$ and rank $1$. A point $P$ of infinite order corresponds to $z=-1/3$. One of the torsion points $T$ corresponds to $z=1/4$ and this value in fact satisfies the starting three conditions. However, the first factor is zero. Other solutions of the system correspond to the points of the form $T+2kP$. For $T+2P$ we get $z=41/100$, hence the solution mentioned in the comment $y=\sqrt{41}/10$. For $T+4P$ we get $z=38904227209/31543891236$ and hence $y=\sqrt{38904227209}/177606$. 


