Suppose that there is a radical combination $a\sqrt{b}+c\sqrt{d}$ where a,b,c,d are natural. 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 $x\sqrt{z}$ - in this case $\sqrt{19}$?
2) what restrictions would be needed if there is no way to figure this out in the general case?
Note: i lost my unregistered account - so unable to edit, so I posted this again as a separate question. Can anyone close the first one?