a) \(\sqrt{6+2\sqrt{4-2\sqrt{3}}}\)
\(=\sqrt{6+2\sqrt{\left(\sqrt{3}-1\right)^2}}\)
\(=\sqrt{6+2\left(\sqrt{3}-1\right)}\)
\(=\sqrt{6+2\sqrt{3}-2}\)
\(=\sqrt{4+2\sqrt{3}}\)
\(=\sqrt{\left(\sqrt{3}+1\right)^2}\)
\(=\sqrt{3}+1\)
b) \(\sqrt{12-6\sqrt{3}}\)
\(=\sqrt{9-2\cdot3\cdot\sqrt{3}+3}\)
\(=\sqrt{\left(\sqrt{9}-\sqrt{3}\right)^2}\)
\(=3-\sqrt{3}\)
c) \(\sqrt{6-4\sqrt{2}+\sqrt{22-12\sqrt{2}}}\)
\(=\sqrt{6-4\sqrt{2}+\sqrt{\left(\sqrt{18}-2\right)^2}}\)
\(=\sqrt{6-4\sqrt{2}+\sqrt{18}-2}\)
\(=\sqrt{4-4\sqrt{2}+3\sqrt{2}}\)
\(=\sqrt{4-\sqrt{2}}\)
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