Đặt \(A=\sqrt{4+\sqrt{10+2\sqrt{5}}}+\sqrt{4-\sqrt{10+2\sqrt{5}}}\)
\(\Leftrightarrow A^2=4+\sqrt{10+2\sqrt{5}}+4-\sqrt{10+2\sqrt{5}}+2\sqrt{\left(4+\sqrt{10+2\sqrt{5}}\right)\left(4-\sqrt{10+2\sqrt{5}}\right)}\)
\(\Leftrightarrow A^2=8+2\sqrt{16-10-2\sqrt{5}}\)
\(\Leftrightarrow A^2=8+2\sqrt{6-2\sqrt{5}}\)
\(\Leftrightarrow A^2=8+2\sqrt{\left(\sqrt{5}-1\right)^2}\)
\(\Leftrightarrow A^2=8+2\left(\sqrt{5}-1\right)\)
\(\Leftrightarrow A^2=8+2\sqrt{5}-2\)
\(\Leftrightarrow A^2=6+2\sqrt{5}\)
\(\Leftrightarrow A^2=\left(\sqrt{5}+1\right)^2\)
\(\Leftrightarrow A=\sqrt{5}+1\) ( vì \(A>0\) )
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