\(\sqrt{6+2\sqrt{2}.\sqrt{3-\sqrt{4+2\sqrt{3}}}}\)
= \(\sqrt{6+2\sqrt{2}.\sqrt{3-\sqrt{3+2\sqrt{3}.1}+1}}\)
=\(\sqrt{6+2\sqrt{2}.\sqrt{3-\sqrt{\left(\sqrt{3}+1\right)^2}}}\)
=\(\sqrt{6+2\sqrt{2}.\sqrt{3-\left|\sqrt{3}+1\right|}}\)
=\(\sqrt{6+2\sqrt{2}.\sqrt{3-\sqrt{3}-1}}\)
=\(\sqrt{6+2\sqrt{2}.\sqrt{2-\sqrt{3}}}\)
=\(\sqrt{6+2.\left(\sqrt{2}.\sqrt{2-\sqrt{3}}\right)}\)
=\(\sqrt{6+2.\left(\sqrt{4-2\sqrt{3}}\right)}\)
=\(\sqrt{6+2.\sqrt{\left(\sqrt{3}-1\right)^2}}\)
=\(\sqrt{6+2.\left|\sqrt{3}-1\right|}\)
=\(\sqrt{6+2\sqrt{3}-2}\)
=\(\sqrt{4+2\sqrt{3}}\)
=\(\sqrt{\left(\sqrt{3}+1\right)^2}\)
=\(\left|\sqrt{3}+1\right|\)
=\(\sqrt{3}+1\)
