Noting that $6=2\times 3$, $14=2\times 7$, and $21=3\times 7$ gives: \begin{align} &\sqrt{12+2\sqrt{6}+2\sqrt{14}+2\sqrt{21}}\\ &=\sqrt{(\sqrt{2})^2+\sqrt{3})^2+\sqrt{7})^2+2\cdot\sqrt{2}\cdot\sqrt{3}+2\cdot\sqrt{2}\cdot\sqrt{7}+2\cdot\sqrt{3}\cdot\sqrt{7}}\\ &=\sqrt{(\sqrt{2}+\sqrt{3} +\sqrt{7})^2}\\ &=\sqrt{2}+\sqrt{3} +\sqrt{7} \end{align}

Note: handling nested radical expressions is discussed in the book  Power Calculation .