\(P=\sqrt{13+30\sqrt{2+\sqrt{9+4\sqrt{2}}}}\)
\(=\sqrt{13+30\sqrt{2+2\sqrt{2}+1}}\)
\(=\sqrt{13+30\left(\sqrt{2}+1\right)}\)
\(=\sqrt{13+30\sqrt{2}+30}\)
\(=\sqrt{25+2.5.3\sqrt{2}+18}\)
\(=5+3\sqrt{2}\)
\(P=\sqrt{13+30\sqrt{2+\sqrt{9+4\sqrt{2}}}}\)
\(=\sqrt{13+30\sqrt{2+\sqrt{8+4\sqrt{2}+1}}}\)
\(=\sqrt{13+30\sqrt{2+\sqrt{\left(\sqrt{8}+1\right)^2}}}\)
\(=\sqrt{13+30\sqrt{2+\sqrt{8}+1}}\)
\(=\sqrt{13+30\sqrt{3+\sqrt{8}}}\)
\(=\sqrt{13+30\sqrt{3+2\sqrt{2}}}\)
\(=\sqrt{13+30\sqrt{2+2\sqrt{2}+1}}\)
\(=\sqrt{13+30\sqrt{\left(\sqrt{2}+1\right)^2}}\)
\(=\sqrt{13+30\left(\sqrt{2}+1\right)}\)
\(=\sqrt{13+30\sqrt{2}+30}\)
\(=\sqrt{43+30\sqrt{2}}\)
\(=\sqrt{\left(5+3\sqrt{2}\right)^2}\)
\(=5+3\sqrt{2}\)
