\(D=\sqrt{\dfrac{\left(\sqrt{7+\sqrt{5}}+\sqrt{7-\sqrt{5}}\right)^2}{7+2\sqrt{11}}}-\sqrt{3-2\sqrt{2}}\)
\(=\sqrt{\dfrac{14+2\sqrt{49-5}}{7+2\sqrt{11}}}-\sqrt{\left(\sqrt{2}-1\right)^2}\)
\(=\sqrt{\dfrac{2\left(7+2\sqrt{11}\right)}{7+2\sqrt{11}}}-\left|\sqrt{2}-1\right|\)
\(=\sqrt{2}-\left(\sqrt{2}-1\right)\)
\(=1\)
\(E=\sqrt{2+\sqrt{2}-2\sqrt{\sqrt{2}+1}}+1\)
\(=\sqrt{\sqrt{2}+1-2\sqrt{\sqrt{2}+1}+1}+1\)
\(=\sqrt{\left(\sqrt{\sqrt{2}+1}-1\right)^2}+1\)
\(=\left|\sqrt{\sqrt{2}+1}-1\right|+1\)
\(=\sqrt{\sqrt{2}+1}-1+1\)
\(=\sqrt{\sqrt{2}+1}\)
\(F=\sqrt{3+\sqrt{2}-\sqrt{8\sqrt{2}+8}}-\sqrt{\sqrt{2}+1}\)
\(=\sqrt{3+\sqrt{2}-2.\sqrt{2}\sqrt{\sqrt{2}+1}}-\sqrt{\sqrt{2}+1}\)
\(=\sqrt{\sqrt{2}+1-2.\sqrt{\sqrt{2}+1}.\sqrt{2}+2}-\sqrt{\sqrt{2}+1}\)
\(=\sqrt{\left(\sqrt{\sqrt{2}+1}-\sqrt{2}\right)^2}-\sqrt{\sqrt{2}+1}\)
\(=\left|\sqrt{\sqrt{2}+1}-\sqrt{2}\right|-\sqrt{\sqrt{2}+1}\)
\(=\sqrt{\sqrt{2}+1}-\sqrt{2}-\sqrt{\sqrt{2}+1}\)
\(=-\sqrt{2}\)
g.
Chú ý rằng: \(1=5-3\sqrt{2}+3\sqrt{2}-4\)
Và \(\left(5-3\sqrt{2}\right)\left(3\sqrt{2}-4\right)=2^2.\left(27\sqrt{2}-38\right)\)
Nên:
\(G=\dfrac{\sqrt{5-3\sqrt{2}+2\sqrt{5-3\sqrt{2}}.\sqrt{3\sqrt{2}-4}+3\sqrt{2}-4}-\sqrt{5-3\sqrt{2}}}{\sqrt{3\sqrt{2}-4}}\)
\(=\dfrac{\sqrt{\left(\sqrt{5-3\sqrt{2}}+\sqrt{3\sqrt{2}-4}\right)^2}-\sqrt{5-3\sqrt{2}}}{\sqrt{3\sqrt{2}-4}}\)
\(=\dfrac{\sqrt{5-3\sqrt{2}}+\sqrt{3\sqrt{2}-4}-\sqrt{5-3\sqrt{2}}}{\sqrt{3\sqrt{2}-4}}\)
\(=\dfrac{\sqrt{3\sqrt{2}-4}}{\sqrt{3\sqrt{2}-4}}=1\)
