e) \(\sqrt{12+6\sqrt{3}}\)
\(=\sqrt{3^2+2\cdot3\cdot\sqrt{3}+\left(\sqrt{3}\right)^2}\)
\(=\sqrt{\left(3+\sqrt{3}\right)^2}\)
\(=3+\sqrt{3}\)
f) \(\sqrt{6-3\sqrt{3}}+\sqrt{2-\sqrt{3}}\)
\(=\dfrac{\sqrt{2\left(6-3\sqrt{3}\right)}}{\sqrt{2}}+\dfrac{\sqrt{2\left(2-\sqrt{3}\right)}}{\sqrt{2}}\)
\(=\dfrac{\sqrt{12-6\sqrt{3}}}{\sqrt{2}}+\dfrac{\sqrt{4+2\sqrt{3}}}{\sqrt{2}}\)
\(=\dfrac{\sqrt{3^2-2\cdot3\cdot\sqrt{3}+\left(\sqrt{3}\right)^2}}{\sqrt{2}}+\dfrac{\sqrt{\left(\sqrt{3}\right)^2+2\cdot\sqrt{3}\cdot1+1^2}}{\sqrt{2}}\)
\(=\dfrac{\sqrt{\left(3-\sqrt{3}\right)^2}}{\sqrt{2}}+\dfrac{\sqrt{\left(\sqrt{3}+1\right)^2}}{\sqrt{2}}\)
\(=\dfrac{3-\sqrt{3}}{\sqrt{2}}+\dfrac{\sqrt{3}+1}{\sqrt{2}}\)
g) \(\sqrt{11-2\sqrt{30}}+\sqrt{13+2\sqrt{42}}\)
\(=\sqrt{\left(\sqrt{6}\right)^2-2\cdot\sqrt{6}\cdot\sqrt{5}+\left(\sqrt{5}\right)^2}+\sqrt{\left(\sqrt{6}\right)^2+2\cdot\sqrt{6}\cdot\sqrt{7}+\left(\sqrt{7}\right)^2}\)
\(=\sqrt{\left(\sqrt{6}-\sqrt{5}\right)^2}+\sqrt{\left(\sqrt{6}+\sqrt{7}\right)^2}\)
\(=\sqrt{6}-\sqrt{5}+\sqrt{6}+\sqrt{7}\)
\(=2\sqrt{6}-\sqrt{5}+\sqrt{7}\)
h) \(\sqrt{13+30\sqrt{2+\sqrt{9+4\sqrt{2}}}}\)
\(=\sqrt{13+30\sqrt{2+\sqrt{\left(2\sqrt{2}\right)^2+2\cdot2\sqrt{2}\cdot1+1^2}}}\)
\(=\sqrt{13+30\sqrt{2+\sqrt{\left(2\sqrt{2}+1\right)^2}}}\)
\(=\sqrt{13+30\sqrt{2+2\sqrt{2}+1}}\)
\(=\sqrt{13+30\sqrt{3+2\sqrt{2}}}\)
\(=\sqrt{13+30\sqrt{\left(\sqrt{2}\right)^2+2\cdot\sqrt{2}\cdot1+1^2}}\)
\(=\sqrt{13+30\sqrt{\left(\sqrt{2}+1\right)^2}}\)
\(=\sqrt{13+30\left(\sqrt{2}+1\right)}\)
\(=\sqrt{13+30\sqrt{2}+30}\)
\(=\sqrt{43+30\sqrt{2}}\)
\(=\sqrt{\left(3\sqrt{2}\right)^2+2\cdot3\sqrt{2}\cdot5+5^2}\)
\(=\sqrt{\left(3\sqrt{2}+5\right)^2}\)
\(=3\sqrt{2}+5\)
