Question

$\boxed{\color{red}{\text{Nhanh vs ạ :')}}}$

Answer 1

\(\sqrt{13+30\sqrt{2+\sqrt{9+4\sqrt{2}}}}\\ =\sqrt{13+30\sqrt{2+\sqrt{9+2\sqrt{8}}}}\\ =\sqrt{13+30\sqrt{2+\sqrt{8+2\sqrt{2}.1+1}}}\)

\(=\sqrt{13+30\sqrt{2+\sqrt{\left(2\sqrt{2}+1\right)^2}}}\)

\(=\sqrt{13+30\sqrt{2+2\sqrt{2}+1}}\)

\(=\sqrt{13+30\sqrt{\left(\sqrt{2}+1\right)^2}}\)

\(=\sqrt{13+30\left(\sqrt{2}+1\right)}\)

\(=\sqrt{43+30\sqrt{2}}\)

\(=\sqrt{43+2\sqrt{450}}\)

\(=\sqrt{18+2\sqrt{18}\sqrt{25}+25}\)

\(=\sqrt{\left(5+3\sqrt{2}\right)^2}\\ =5+3\sqrt{2}\)