Ta có \(\sqrt{18-\sqrt{128}}\)
= \(\sqrt{18-8\sqrt{2}}\)
= \(\sqrt{16-2×4×\sqrt{2}+2}\)
= \(4-\sqrt{2}\)
Từ đó cái ban đầu
= \(\sqrt{6+2\sqrt{2}\sqrt{3-\sqrt{4+2\sqrt{3}}}}\)
= \(\sqrt{6+2\sqrt{2}\sqrt{2-\sqrt{3}}}\)
= \(\sqrt{6+2\sqrt{4-2\sqrt{3}}}\)
= \(\sqrt{6+2\sqrt{3}-2}\)
= \(\sqrt{4+2\sqrt{3}}\)
= \(\sqrt{3}+1\)
Ta có: \(\sqrt{6+2\sqrt{2}.\sqrt{3-\sqrt{\sqrt{2}+\sqrt{12}+\sqrt{18-\sqrt{128}}}}}\)
\(=\sqrt{6+2\sqrt{2}.\sqrt{3-\sqrt{\sqrt{2}+\sqrt{12}+\sqrt{16-8\sqrt{2}+2}}}}\)
\(=\sqrt{6+2\sqrt{2}.\sqrt{3-\sqrt{\sqrt{2}+\sqrt{12}+\sqrt{\left(4-\sqrt{2}\right)^2}}}}\)
\(=\sqrt{6+2\sqrt{2}.\sqrt{3-\sqrt{\sqrt{2}+\sqrt{12}+4-\sqrt{2}}}}\)
\(=\sqrt{6+2\sqrt{2}.\sqrt{3-\sqrt{3+2\sqrt{3}+1}}}\)
\(=\sqrt{6+2\sqrt{2}.\sqrt{3-\sqrt{\left(\sqrt{3}+1\right)^2}}}\)
\(=\sqrt{6+2\sqrt{2}.\sqrt{3-\sqrt{3}+1}}\)
\(=\sqrt{6+2.\sqrt{4-2\sqrt{3}}}\)
\(=\sqrt{6+2.\sqrt{3-2\sqrt{3}+1}}\)
\(=\sqrt{6+2.\sqrt{\left(\sqrt{3}-1\right)^2}}\)
\(=\sqrt{6+2.\sqrt{3}-2}\)
\(=\sqrt{3+2\sqrt{3}+1}\)
\(=\sqrt{\left(\sqrt{3}+1\right)^2}\)
\(=\sqrt{3}+1\)
