Suppose that there is a radical combination $a\sqrt{b}+c\sqrt{d} ...$. where $a,b,c,d\in \mathbb{N}$, for which each term part $\sqrt{b}$ cannot be transformed into the form of $s\sqrt{q}$.
The question is,
1) Suppose that we convert this combination into a rational number approximation. Is there any quick way to know the number of terms that cannot or can be reduced to the form of $x\sqrt{z}$ in the original square root combination using an approximate value? (This would mean that an approximate value would be unique to a particular combination.)
Edit: For example, $12\sqrt{13} + 15\sqrt{17} + \sqrt{19}$. We do addition operation and convert it into a decimal approximation. Using the approximation value how would we be able to know the term that is not of form $x\sqrt{z}$?
2) What restrictions would be needed if there is no way to figure this out in the general case?