Ta có : \(\sqrt{4+\sqrt{10+2\sqrt{5}}}+\sqrt{4-\sqrt{10+2\sqrt{5}}}\)
Ta thấy : \(\sqrt{4+\sqrt{10+2\sqrt{5}}}+\sqrt{4-\sqrt{10+2\sqrt{5}}}\ge0\)
= \(\sqrt{\left(\sqrt{4+\sqrt{10+2\sqrt{5}}}+\sqrt{4-\sqrt{10+2\sqrt{5}}}\right)}^2\)
\(=\sqrt{4+\sqrt{10+2\sqrt{5}}+4-\sqrt{10+2\sqrt{5}}+2\sqrt{\left(4+\sqrt{10+2\sqrt{5}}\right)\left(4-\sqrt{10+2\sqrt{5}}\right)}}\)
\(=\sqrt{8+2\sqrt{16-10-2\sqrt{5}}}=\sqrt{8+2\sqrt{6-2\sqrt{5}}}\)
\(=\sqrt{8+2\sqrt{5-2\sqrt{5}+1}}=\sqrt{8+2\sqrt{\left(\sqrt{5}-1\right)^2}}\)
\(=\sqrt{8+2\sqrt{5}-2}=\sqrt{6+2\sqrt{5}}=\sqrt{5+2\sqrt{5}+1}\)
\(=\sqrt{\left(\sqrt{5}+1\right)^2}=1+\sqrt{5}\)
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