$\sqrt{11+4\sqrt{7}}=\sqrt{7+4+4\sqrt{7}}=\sqrt{(\sqrt{7}{)}^2+2\ast <!--\; \ast \; -->2\sqrt{7}+2^2}=\sqrt{(\sqrt{7}+2{)}^2}=|\sqrt{7}+2|=\sqrt{7}+2$

$\sqrt{14+8\sqrt{3}}=\sqrt{8+6+2\ast <!--\; \ast \; -->4\sqrt{3}}=\sqrt{(\sqrt{8}{)}^2+2\sqrt{16\ast <!--\; \ast \; -->3}+(\sqrt{6}{)}^2}=\sqrt{(\sqrt{8}{)}^2+2\sqrt{48}+(\sqrt{6}{)}^2}=\sqrt{(\sqrt{8}{)}^2+2\sqrt{8\ast <!--\; \ast \; -->6}+(\sqrt{6}{)}^2}=\sqrt{(\sqrt{8}+\sqrt{6}{)}^2}=|\sqrt{8}+\sqrt{6}|=\sqrt{8}+\sqrt{6}$