Let $a,b,c,d,e$ be positive integers, and suppose it is simple to compare integers to each other, but difficult to compare irrationals, and we want to compare sums of square roots.
As a simple example, we can show that $\sqrt{a}+\sqrt{b}>\sqrt{c}$ is equivalent to either of $a+b>c$ or $4ab>(c-a-b)^2$ being true, by some simple algebra and being careful of signs. So here we can reduce a comparison of sums of square roots to a comparison of integers.
The next case of interest would be comparing 4 radicals in something like $\sqrt{a}+\sqrt{b}>\sqrt{c}+\sqrt{d}$, which with a bit more squaring and keeping track of different cases, can still be reduced to an equivalent set of integer comparisons.
However, when there are 5 radicals in the comparison (e.g. $\sqrt{a}+\sqrt{b}>\sqrt{c}+\sqrt{d}+\sqrt{e}$), I am unable to make much progress to reducing the number of radicals, and I'm not sure if it is even possible.
Is there an easier way to compare such sums of square roots in general, and are there always equivalent integer comparisons? If instead I approximate the square roots by some numerical method, how much accuracy is required for guaranteed correctness?