TITLE: About rationalizing expressions
QUESTION [7 upvotes]: For example, rationalizing expressions like
$$\frac{1}{\pm \sqrt{a} \pm \sqrt{b}}$$
Is straightforward. Moreover cases like 
$$\frac{1}{\pm \sqrt{a} \pm \sqrt{b} \pm \sqrt{c}}$$
and
$$\frac{1}{\pm \sqrt{a} \pm \sqrt{b} \pm \sqrt{c} \pm \sqrt{d}}$$
Are still easy to rationalize. But my question is in the more general case
$$\frac{1}{\pm \sqrt{a_1} \pm \sqrt{a_2} \cdots \pm \sqrt{a_n}}$$
Where $n \ge 5$
Are they always rationalizable? If so, how would be an algorithm to rationalize them. If not, then a proof must exist.
From my point of view, I can't find an obvious way to rationalize the case $n=5$, since grouping the radicals in a group of 3 and a group of 2 radicals and then applying the identity
$$(a-b)*(a+b)=a^2-b^2$$
Just modifies the denominator from
$$\pm \sqrt{a} \pm \sqrt{b} \pm \sqrt{c} \pm \sqrt{d} \pm \sqrt{e}$$
to
$$\pm v \pm \sqrt{w} \pm \sqrt{x} \pm \sqrt{y} \pm \sqrt{z}$$
Will this help in something? Or a different method or identity is needed?

REPLY [4 votes]: 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 $\rho \in \{-1,1\}^n$,
$$ \frac{1}{\sum_i \rho_i \sqrt{a_i}} = \dfrac{\prod_{\sigma \ne \rho} \sum_i \sigma_i \sqrt{a_i}}{P_n(\sqrt{a_1},\ldots,\sqrt{a_n})}$$