\(\frac{\sqrt{1+\frac{2\sqrt{2}}{3}}+\sqrt{1-\frac{2\sqrt{2}}{3}}}{\sqrt{1+\frac{2\sqrt{2}}{3}}-\sqrt{1-\frac{2\sqrt{2}}{3}}}=\frac{\sqrt{\frac{3+2\sqrt{2}}{3}}+\sqrt{\frac{3-2\sqrt{2}}{3}}}{\sqrt{\frac{3+2\sqrt{2}}{3}}-\sqrt{\frac{3-2\sqrt{2}}{3}}}=\frac{\frac{\sqrt{\left(1+\sqrt{2}\right)^2}+\sqrt{\left(\sqrt{2}-1\right)^2}}{\sqrt{3}}}{\frac{\sqrt{\left(1+\sqrt{2}\right)^2}-\sqrt{\left(\sqrt{2}-1\right)^2}}{\sqrt{3}}}\)\(=\frac{1+\sqrt{2}+\sqrt{2}-1}{1+\sqrt{2}-\sqrt{2}+1}=\frac{2\sqrt{2}}{2}=\sqrt{2}\left(đpcm\right)\)
\(VT=\frac{\left(\sqrt{1+\frac{2\sqrt{2}}{3}}+\sqrt{1-\frac{2\sqrt{2}}{3}}\right)^2}{\left(\sqrt{1+\frac{2\sqrt{2}}{3}}+\sqrt{1-\frac{2\sqrt{2}}{3}}\right)\left(\sqrt{1+\frac{2\sqrt{2}}{3}}-\sqrt{1-\frac{2\sqrt{2}}{3}}\right)}\)
\(=\frac{1+\frac{2\sqrt{2}}{3}+1-\frac{2\sqrt{2}}{3}+2\sqrt{\left(1+\frac{2\sqrt{2}}{3}\right)\left(1-\frac{2\sqrt{2}}{3}\right)}}{1+\frac{2\sqrt{2}}{3}-\left(1-\frac{2\sqrt{2}}{3}\right)}\)
= \(=\frac{2+2\sqrt{1-\frac{8}{9}}}{1+\frac{2\sqrt{2}}{3}-1+\frac{2\sqrt{2}}{3}}\)
\(=\frac{2+2\cdot\frac{1}{3}}{\frac{4\sqrt{2}}{3}}=\frac{\frac{8}{3}}{\frac{4\sqrt{2}}{3}}=\frac{8}{3}\cdot\frac{3}{4\sqrt{2}}=\frac{2}{\sqrt{2}}=\sqrt{2}=vp\)
=> ĐPCM
