a, Xét \(M^2=4-\sqrt{10-2\sqrt{5}}+4+\sqrt{10-2\sqrt{5}}-2\sqrt{\left(4-\sqrt{10-2\sqrt{5}}\right)\left(4+\sqrt{10-2\sqrt{5}}\right)}\)
\(=8-2\sqrt{4^2-10+2\sqrt{5}}\\ =8-2\sqrt{16-10+2\sqrt{5}}\\ =8-2\sqrt{6+2\sqrt{5}}\\ =8-2\sqrt{\left(\sqrt{5}+1\right)^2}\\ =8-2\left(\sqrt{5}+1\right)\\ =8-2\sqrt{5}-2=6-2\sqrt{5}=\left(\sqrt{5}-1\right)^2\\ \Rightarrow M=\sqrt{\left(\sqrt{5}-1\right)^2}=\sqrt{5}-1\)
b,
\(P\sqrt{2}=\frac{\sqrt{2}\left(2+\sqrt{3}\right)}{2+\sqrt{4+2\sqrt{3}}}+\frac{\sqrt{2}\left(2-\sqrt{3}\right)}{2-\sqrt{4-2\sqrt{3}}}\\ =\frac{\sqrt{2}\left(2+\sqrt{3}\right)}{2+\sqrt{\left(\sqrt{3}+1\right)^2}}+\frac{\sqrt{2}\left(2-\sqrt{3}\right)}{2-\sqrt{\left(\sqrt{3}-1\right)^2}}\\ =\frac{\sqrt{2}\left(2+\sqrt{3}\right)}{2+\sqrt{3}+1}+\frac{\sqrt{2}\left(2-\sqrt{3}\right)}{2-\sqrt{3}+1}\\ =\frac{\sqrt{2}\left(2+\sqrt{3}\right)}{3+\sqrt{3}}+\frac{\sqrt{2}\left(2-\sqrt{3}\right)}{3-\sqrt{3}}\\ =\frac{\sqrt{2}\left(2+\sqrt{3}\right)}{\sqrt{3}\left(\sqrt{3}+1\right)}+\frac{\sqrt{2}\left(2-\sqrt{3}\right)}{\sqrt{3}\left(\sqrt{3}-1\right)}\\ =\frac{\sqrt{2}\left[\left(2+\sqrt{3}\right)\left(\sqrt{3}-1\right)+\left(2-\sqrt{3}\right)\left(\sqrt{3}+1\right)\right]}{\sqrt{3}\left(\sqrt{3}+1\right)\left(\sqrt{3}-1\right)}\\ =\frac{\sqrt{2}\left(2\sqrt{3}+3-2-\sqrt{3}+2\sqrt{3}-3+2-\sqrt{3}\right)}{\sqrt{3}\left(3-1\right)}\\ =\frac{\sqrt{2}\left(2\sqrt{3}\right)}{\sqrt{3}\cdot2}=\sqrt{2}\\ \Rightarrow P=1\)
c,
\(Q\sqrt{2}=\sqrt{4-2\sqrt{3}}+\sqrt{4+2\sqrt{3}}\\ =\sqrt{\left(\sqrt{3}-1\right)^2}+\sqrt{\left(\sqrt{3}+1\right)^2}\\ =\sqrt{3}-1+\sqrt{3}+1=2\sqrt{3}\\ \Rightarrow Q=\frac{2\sqrt{3}}{\sqrt{2}}=\sqrt{2}\cdot\sqrt{3}=6\)
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