\(M=\sqrt{3-\sqrt{5}}-\sqrt{3+\sqrt{5}}\)
\(=\dfrac{\sqrt{6-2\sqrt{5}}-\sqrt{6+2\sqrt{5}}}{\sqrt{2}}\)
\(=\dfrac{\sqrt{5}-1-\sqrt{5}-1}{\sqrt{2}}\)
\(=-\sqrt{2}\)
\(M=\sqrt{3-\sqrt{5}}-\sqrt{3+\sqrt{5}}=\sqrt{\left(\sqrt{\dfrac{5}{2}}-\sqrt{\dfrac{1}{2}}\right)^2}-\sqrt{\left(\sqrt{\dfrac{5}{2}}+\sqrt{\dfrac{1}{2}}\right)^2}=\sqrt{\dfrac{5}{2}}-\sqrt{\dfrac{1}{2}}-\sqrt{\dfrac{5}{2}}-\sqrt{\dfrac{1}{2}}=-2\sqrt{\dfrac{1}{2}}=-\sqrt{2}\)
\(M^2=3-\sqrt{5}+3+\sqrt{5}-2\sqrt{\left(3-\sqrt{5}\right)\left(3+\sqrt{5}\right)}\\ M^2=6-2\sqrt{1}=4\\ \Leftrightarrow\left[{}\begin{matrix}M=2\\M=-2\end{matrix}\right.\)
\(\Rightarrow\sqrt{2}.M=\sqrt{6-2\sqrt{5}}-\sqrt{6+2\sqrt{5}}\)
\(\Rightarrow\sqrt{2}.M=\sqrt{\left(\sqrt{5}-1\right)^2}-\sqrt{\left(\sqrt{5}+1\right)^2}\)
\(\Rightarrow\sqrt{2}.M=\sqrt{5}-1-\sqrt{5}-1=-2\)
\(\Rightarrow M=\dfrac{-2}{\sqrt{2}}\)
