<s>[INST] Evaluate $(\sqrt{5}+\sqrt{6}+\sqrt{7})(\sqrt{5}+\sqrt{6}-\sqrt{7})(\sqrt{5}-\sqrt{6}+\sqrt{7})(-\sqrt{5}+\sqrt{6}+\sqrt{7})$ Evaluate $(\sqrt{5}+\sqrt{6}+\sqrt{7})(\sqrt{5}+\sqrt{6}-\sqrt{7})(\sqrt{5}-\sqrt{6}+\sqrt{7})(-\sqrt{5}+\sqrt{6}+\sqrt{7})$
I find out the answer $104$ but my friend find the answer $96$ using calculator.which one is correct?
 [/INST] Generally:
$$\begin{aligned}
&\phantom{mm}(a+b+c)(a+b-c)(a-b+c)(-a+b+c)\\
&=((a+b)+c)((a+b)-c) \times ( c+(a-b))(c-(a-b)) \\
&=((a+b)^2 -c^2)(c^2-(a-b)^2)\\
&=[2ab+(a^2+b^2 - c^2) ] [ 2ab - (a^2+b^2-c^2)]\\
&= 4a^2b^2 - (a^2+b^2-c^2)^2.
\end{aligned}$$
Here $a^2=5, b^2=6,$ and $c^2=7$, so the answer is $104$ as you said.
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