\sqrt{2}*\sqrt{32}=\sqrt{2*32}=\sqrt{64}=8
\sqrt[4]{80}:\sqrt[4]{5}=\sqrt[4]{5*16}:\sqrt[4]{5}=\sqrt[4]{5}*\sqrt[4]{16}:\sqrt[4]{5}=\sqrt[4]{16}=2
\frac{7}{\sqrt{2}+3}+1=\frac{7(\sqrt{2}-3)}{(\sqrt{2}+3)(\sqrt{2}+3)}+1=\frac{7(\sqrt{2}-3)}{(\sqrt{2})^2-3^2}+1=\frac{7(\sqrt{2}-3)}{2-9}+1=\frac{7(\sqrt{2}-3)}{-7}+1=-\sqrt{2}+3+1=-\sqrt{2}+4=4-\sqrt{2}