\(\sqrt{13+\sqrt{30+\sqrt{2+\sqrt{9+4\sqrt{2}}}}}\)
\(=\sqrt{13+30\sqrt{3+2\sqrt{2}}}\)
\(=\sqrt{13+30\left(\sqrt{2}+1\right)}\)
\(=\sqrt{43+30\sqrt{2}}\)
\(=5+3\sqrt{2}\)
\(=\sqrt{13+\sqrt{30+\sqrt{2+\sqrt{8+2\cdot2\sqrt{2}+1}}}}=\sqrt{13+\sqrt{30+\sqrt{2+\sqrt{\left(2\sqrt{2}\right)^2+2\cdot2\sqrt{2}+1}}}}=\sqrt{13+\sqrt{30+\sqrt{2+\sqrt{\left(2\sqrt{2}\text{+}1\right)^2}}}}=\sqrt{13+\sqrt{30\text{+}\sqrt{2\text{+}2\sqrt{2}+1}}}=\sqrt{13+\sqrt{30+\sqrt{\left(\sqrt{2}+1\right)^2}}}=\sqrt{13+\sqrt{30+\sqrt{2}+1}}=\sqrt{13+\sqrt{31+\sqrt{2}}}\)
