What is \((\sqrt{2} + \sqrt{3} + \sqrt{5})(\sqrt{2} + \sqrt{3} - \sqrt{5})(\sqrt{2} - \sqrt{3} + \sqrt{5})(-\sqrt{2} + \sqrt{3} + \sqrt{5})\) equal to?
24
Let \(a=\sqrt{2}, b=\sqrt{3}, c=\sqrt{5}\). Group the factors as \((a+b+c)(a+b-c)\) and \((a-b+c)(-a+b+c)\).
\((a+b+c)(a+b-c)=(a+b)^2-c^2=(5+2\sqrt{6})-5=2\sqrt{6}\).
\((a-b+c)(-a+b+c)=c^2-(a-b)^2=5-(5-2\sqrt{6})=2\sqrt{6}\).
So the product \(=2\sqrt{6}\times 2\sqrt{6}=4\times 6=24\).
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