\(M=\sqrt{4+\sqrt{10+2\sqrt{5}}}+\sqrt{4-\sqrt{10+2\sqrt{5}}}\)
\(\Rightarrow M^2=\left(4+\sqrt{10+2\sqrt{5}}\right)+2\sqrt{\left(\sqrt{4+\sqrt{10+2\sqrt{5}}}\right)\left(\sqrt{4-\sqrt{10+2\sqrt{5}}}\right)}+\left(4-\sqrt{10+2\sqrt{5}}\right)\)
\(=8+2\sqrt{16-\left(10+2\sqrt{5}\right)}\)
\(=8+2\sqrt{\left(\sqrt{5}-1\right)^2}\)
\(=8+2\sqrt{5}-2\)
\(=\left(\sqrt{5}+1\right)^2\)
\(\Rightarrow\sqrt{M^2}=\sqrt{\left(\sqrt{5}+1\right)^2}\)
\(\Rightarrow\left|M\right|=\sqrt{5}+1\) mà M > 0
\(\Rightarrow M=\sqrt{5}+1\)
\(M=\sqrt{4+\sqrt{15}}+\sqrt{4-\sqrt{15}}-2\sqrt{3-\sqrt{5}}\)
\(\Rightarrow\sqrt{2}M=\sqrt{8+2\sqrt{15}}+\sqrt{8-2\sqrt{15}}-2\sqrt{6-2\sqrt{5}}\)
\(=\sqrt{\left(\sqrt{5}+\sqrt{3}\right)^2}+\sqrt{\left(\sqrt{5}-\sqrt{3}\right)^2}-2\sqrt{\left(\sqrt{5}-1\right)^2}\)
\(=\sqrt{5}+\sqrt{3}+\sqrt{5}-\sqrt{3}-2\sqrt{5}+2\)
= 2
\(\Rightarrow M=\sqrt{2}\)
