<s>[INST] Multiple radicals: $(\sqrt{10}+\sqrt{11}+\sqrt{12})(\sqrt{10}+\sqrt{11}-\sqrt{12})(\sqrt{10}-\sqrt{11}+\sqrt{12})(-\sqrt{10}+\sqrt{11}+\sqrt{12})$ $(\sqrt{10}+\sqrt{11}+\sqrt{12})(\sqrt{10}+\sqrt{11}-\sqrt{12})(\sqrt{10}-\sqrt{11}+\sqrt{12})(-\sqrt{10}+\sqrt{11}+\sqrt{12})$
I don't think multiplying these out will work, and I am stuck in the beginning, without a basic concept to get started. Can anyone show me how to do these? I would appreciate more detailed responses as I have a solution to this already but since it does not explain the steps, I cannot learn from it. Thanks



Answer, if you want to check

 359

 [/INST] HINT:
$$(a+b+c)(a+b-c)(b+c-a)(c+a-b)$$
$$=\{(a+b)^2-c^2\}\{c^2-(a-b)^2\}$$
$$=-(a^2+b^2+2ab-c^2)(a^2+b^2-2ab-c^2)$$
$$=-\{(a^2+b^2-c^2)^2-(2ab)^2\}$$
which is symmetric on expansion.
So, WLOG choose $\{a,b,c\}$ from $\{\sqrt{10},\sqrt{11},\sqrt{12}\}$ to find the same result.
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