$\sqrt{13+30\sqrt{2+\sqrt{9+4\sqrt{2}}}}$
$= \sqrt{13+30\sqrt{2+\sqrt{8+4\sqrt{2}+1}}}$
$= \sqrt{13+30\sqrt{2+\sqrt{(2\sqrt{2}+1)²}}}$
$= \sqrt{13+30\sqrt{2+2\sqrt{2}+1}}$
$= \sqrt{13+30\sqrt{(\sqrt{2}+1)²}}$
$= \sqrt{13+30.(\sqrt{2}+1)}$
$= \sqrt{13+30\sqrt{2}+30}$
$= \sqrt{18+30\sqrt{2}+25}$
$= \sqrt{(3\sqrt{2}+5)²}$
$= 3\sqrt{2}+5$