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