About rationalizing expressions

Consider $$P_n(t_1,\ldots,t_n) = \prod_{\sigma \in \{-1,1\}^n}\sum_i \sigma_i t_i$$ You probably don't want to expand this explicitly for symbolic $t_i$, but it is a polynomial in the $t_i$ whose terms involve only even powers (because it's invariant under $t_i \to -t_i$). Thus $P_n(\sqrt{a_1},\ldots,\sqrt{a_n})$ is a polynomial in $a_1, \ldots, a_n$. Then for any $\rho \in \{-1,1\}^n$, $$ \frac{1}{\sum_i \rho_i \sqrt{a_i}} = \dfrac{\prod_{\sigma \ne \rho} \sum_i \sigma_i \sqrt{a_i}}{P_n(\sqrt{a_1},\ldots,\sqrt{a_n})}$$

Yes, it's always possible to rationalize such expressions.

Without loss of generality, it's enough to consider the first square root having positive sign in front of it. Then, see what happens if you multiply together terms with all possible combinations of the remaining signs:

$$\left(\sqrt{a}+\sqrt{b}\right)\left(\sqrt{a}-\sqrt{b}\right)=a-b$$

$$\begin{eqnarray} \left(\sqrt{a}+\sqrt{b}+\sqrt{c}\right) \left(\sqrt{a}+\sqrt{b}-\sqrt{c}\right)\\ \left(\sqrt{a}-\sqrt{b}+\sqrt{c}\right) \left(\sqrt{a}-\sqrt{b}-\sqrt{c}\right) & = & a^2 + b^2 + c^2 - 2ab - 2ac - 2bc \\ \end{eqnarray}$$

The resulting expressions get progressively more complicated (the resulting polynomial will be homogeneous, symmetric polynomial of degree $2^{n-2}$ -- it'll contain all possible combinations of the variables with all possible combinations of exponents), but it won't contain any of the square roots anymore.