Is any combination of roots (eg, $\sqrt2+\sqrt3$) a unique number?

Question

Every non negative real number has a unique non negative root, called the principle square root. Does this mean that any combination of roots added will also produce a unique value?

For example, $$\sqrt{2}+\sqrt{3} = 1.414213562373095\ldots + 1.732050807568877\ldots = 3.146264369941972\ldots$$ Is this resulting value as unique as each of the two principle square roots that were added?

Is there a way to prove this uniqueness mathematically or logically without producing a huge table of all possible combinations to check against each other?

Answer 1

$\sqrt{2}$ and $\sqrt{3}$ are linearly independent over $\mathbb{Q}$. See here

Answer 2

Since you are assuming the existence and uniqueness of roots of non-negative real numbers (in fact, this is not hard to prove rigorously), Yes. Because $\sqrt2+\sqrt3$ is the principle square root of $5+\sqrt{24}.$