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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?

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    It's quite hard to decipher what you mean. Are you just saying that the sum of the two radicals is not equal to some third number of the same form, i.e. $e\sqrt{f}$ where $e,f$ are positive integers? Without knowing what you mean in the first paragraph there's not much hope of answering the questions you put.2012-10-25
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    I guess the first sentence just wants to say that each number under a radical sign is square-free.2013-08-18

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