Question

\(\sqrt{4+\sqrt{10+2\sqrt{5}}}+\sqrt{4-\sqrt{\sqrt{10+2\sqrt{5}}}}\)

Answer 1

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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