how to rationalize $\frac{1}{\sqrt a+\sqrt b+\sqrt c+\sqrt d+e}$ let say $a, b, c, d, e \in \Bbb Z^+ $ and $\sqrt a, \sqrt b, \sqrt c, \sqrt d \notin \Bbb Z^+ $ My problem is how to rationalize the denominator of $\frac{1}{\sqrt a+\sqrt b+\sqrt c+\sqrt d+e}$ is this can be rationalized? what is the "term" to multiply both the numerator and the denominator? I tried to rationaliz by multiplying $\sqrt a+\sqrt b+\sqrt c-\sqrt d+e$ and another time by multiplying $\sqrt a+\sqrt b-\sqrt c-\sqrt d+e$ but didn't successes can you tell how to rationaliz or a hint also would appreciate Thanks.

In general you have that, $$\Pi_{(i_1,\dots, i_n)\in C_2^n}((-1)^{i_1}\sqrt{a_1}+\dots+(-1)^{i_n}\sqrt{a_n})$$ is an integer,

where $a_i$ is an integer $\forall i$.

Can you prove this?

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I think an elementary proof may be this:

The polynomial $$\Pi_{(i_1,\dots, i_n)\in C_2^n} ((-1)^{i_1}X_1+\dots+(-1)^{i_n}X_n)$$ is clearly even on each of its variables, then it must be quadratic on each of its variables (meaning that it can be rewritten as a polynomial on $X_i^2$). And this concludes.