\(\sqrt{\sqrt{2}+2\sqrt{3}+\sqrt{18-8\sqrt{2}}}\)
\(=\sqrt{\sqrt{2}+2\sqrt{3}+\sqrt{16-8\sqrt{2}+2}}\)
\(=\sqrt{\sqrt{2}+2\sqrt{3}+\sqrt{4^2-2.4.\sqrt{2}+\sqrt{2}^2}}\)
\(=\sqrt{\sqrt{2}+2\sqrt{3}+\sqrt{\left(4-\sqrt{2}\right)^2}}\)
\(=\sqrt{\sqrt{2}+2\sqrt{3}+4-\sqrt{2}}\)
\(=\sqrt{2\sqrt{3}+4}\)
\(=\sqrt{1+2\sqrt{3}+3}\)
\(=\sqrt{1+2\sqrt{3}+\sqrt{3}^2}\)
\(=\sqrt{\left(1+\sqrt{3}\right)^2}\)
\(=1+\sqrt{3}\)
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