Lời giải:
Ta có:
\(5,5+3\sqrt{2}=5,5+2\sqrt{\frac{9}{2}}=\frac{9}{2}+1+2\sqrt{\frac{9}{2}.1}=(\sqrt{\frac{9}{2}}+1)^2\)
\(\Rightarrow \sqrt{5,5+3\sqrt{2}}=\sqrt{\frac{9}{2}}+1\)
Tương tự:\(\sqrt{5,5-3\sqrt{2}}=\sqrt{\frac{9}{2}}-1\)
Do đó:
\(\frac{\sqrt{5,5+3\sqrt{2}}-\sqrt{5,5-3\sqrt{2}}}{6\sqrt{2}}=\frac{\sqrt{\frac{9}{2}}+1-(\sqrt{\frac{9}{2}}-1)}{6\sqrt{2}}\)
\(=\frac{2}{6\sqrt{2}}=\frac{1}{3\sqrt{2}}\)
\(\dfrac{\sqrt{5,5+3\sqrt{2}}-\sqrt{5,5-3\sqrt{2}}}{6\sqrt{2}}=\dfrac{\sqrt{2}\cdot\sqrt{5,5+3\sqrt{2}}-\sqrt{2}\cdot\sqrt{5,5-3\sqrt{2}}}{12}=\dfrac{\sqrt{11+2\cdot3\sqrt{2}}-\sqrt{11-2\cdot3\sqrt{2}}}{12}=\dfrac{\sqrt{9+2\cdot3\cdot\sqrt{2}+2}-\sqrt{9-2\cdot3\cdot\sqrt{2}+2}}{12}=\dfrac{\sqrt{\left(3+\sqrt{2}\right)^2}-\sqrt{\left(3-\sqrt{2}\right)^2}}{12}=\dfrac{3+\sqrt{2}-3+\sqrt{2}}{12}=\dfrac{2\sqrt{2}}{12}=\dfrac{\sqrt{2}}{6}\)
