Question

\(\sqrt{53-20\sqrt{4-\sqrt{9+4\sqrt{2}}}}\)

Answer 1

\(\sqrt{53-20\sqrt{4-\sqrt{9+4\sqrt{2}}}}=\sqrt{53-20\sqrt{4-\sqrt{\left(2\sqrt{2}\right)^2+2.2\sqrt{2}.1+1}}}=\sqrt{53-20\sqrt{4-\sqrt{\left(2\sqrt{2}+1\right)^2}}}=\sqrt{53-20\sqrt{4-2\sqrt{2}-1}}=\sqrt{53-20\sqrt{2-2\sqrt{2}+1}}=\sqrt{53-20\sqrt{\left(\sqrt{2}-1\right)^2}}=\sqrt{53-20\left(\sqrt{2}-1\right)}=\sqrt{53-20\sqrt{2}-20}=\sqrt{25-2.5.2\sqrt{2}+8}=\sqrt{\left(5-2\sqrt{2}\right)^2}=5-2\sqrt{2}\)