Converting an equation based on square roots

Well, you may have to do a lot of squaring, and it may not be practical to do it by hand, but here's the theory: let's start with $\sqrt u+\sqrt v+\sqrt w+\sqrt x+\sqrt y+\sqrt z=k$. Square both sides, transfer all the terms without square roots to the right, divide by two, and you get $\sqrt{uv}+\cdots+\sqrt{yz}=k^2+f(u,\dots,z)$ for some polynomial $f$. Square again and move non-roots to the right. On the left, you get a sum with terms of the type $\sqrt{uv}$ and $\sqrt{uvwx}$, on the right some new polynomial $g(u,\dots,z)$. Do it again, on the left you'll have terms of the type $\sqrt{uv}$, $\sqrt{uvwx}$, and $\sqrt{uvwxyz}$, on the right some polynomial $h(u,\dots,z)$.

Keep on doing this. You'll only ever get terms of those three types on the left, and polynomials on the right. Now there are only $15$ different terms of type $\sqrt{uv}$, another $15$ of type $\sqrt{uvwx}$, and just one of type $\sqrt{uvwxyz}$, making $31$ different terms in all. So after you've done the procedure $32$ times, you'll have $32$ linear equations in these $31$ terms, and you can use linear algebra to boil them down to a single equation with no square roots in it, and you win.

I hope you won't expect me to actually carry out this procedure here....