\(C=\sqrt{5-2\sqrt{3}-2\sqrt{4-2\sqrt{3}}}\\ =\sqrt{5-2\sqrt{3}-2\sqrt{\left(\sqrt{3}\right)^2-2\cdot\sqrt{3}\cdot1+1^2}}\\ =\sqrt{5-2\sqrt{3}-2\sqrt{\left(\sqrt{3}-1\right)^2}}\\ =\sqrt{5-2\sqrt{3}-2\sqrt{3}+2}\\ =\sqrt{7-4\sqrt{3}}\\ =\sqrt{2^2-2\cdot2\cdot\sqrt{3}+\left(\sqrt{3}\right)^2}\\ =\sqrt{\left(2-\sqrt{3}\right)^2}\\ =2-\sqrt{3}\\ C=\sqrt{\sqrt{2}+2\sqrt{3}+\sqrt{18-8\sqrt{2}}}\\ =\sqrt{\sqrt{2}+2\sqrt{3}+\sqrt{4^2-2\cdot4\cdot\sqrt{2}+\left(\sqrt{2}\right)^2}}\\ =\sqrt{\sqrt{2}+2\sqrt{3}+\sqrt{\left(4-\sqrt{2}\right)^2}}\\ =\sqrt{\sqrt{2}+2\sqrt{3}+4-\sqrt{2}}\\=\sqrt{4+2\sqrt{3}}\\ =\sqrt{\left(\sqrt{3}\right)^2+2\cdot\sqrt{3}\cdot1+1^2}\\ =\sqrt{\left(\sqrt{3}+1\right)^2}\\ =\sqrt{3}+1\)
\(C=\sqrt{5-2\sqrt{3}-2\sqrt{4-2\sqrt{3}}}\)
\(\Leftrightarrow C=\sqrt{5-2\sqrt{3}-2\sqrt{3-2\sqrt{3}+1}}\)
\(\Leftrightarrow C=\sqrt{5-2\sqrt{3}-2\sqrt{\left(\sqrt{3}-1\right)^2}}\)
\(\Leftrightarrow C=\sqrt{5-2\sqrt{3}-2\left|\sqrt{3}-1\right|}\)
\(\Leftrightarrow C=\sqrt{5-2\sqrt{3}-2\left(\sqrt{3}-1\right)}\)
\(\Leftrightarrow C=\sqrt{5-2\sqrt{3}-2\sqrt{3}+2}\)
\(\Leftrightarrow C=\sqrt{7-4\sqrt{3}}\)
\(\Leftrightarrow C=\sqrt{4-4\sqrt{3}+3}\)
\(\Leftrightarrow C=\sqrt{\left(2-\sqrt{3}\right)^2}=\left|2-\sqrt{3}\right|=2-\sqrt{3}\)
