Question

a) \(\sqrt{17-4\sqrt{9+4\sqrt{5}}}\)

b)\(\sqrt{\left(\sqrt{2-3}\right)^2}.\sqrt{11+6\sqrt{2}}\)

c) \(\sqrt{\left(\sqrt{3-3}\right)^2.}\sqrt{\frac{1}{3-\sqrt{3}}}\)

d)\(\left(\sqrt{6}-3\sqrt{3}+5\sqrt{2}-\frac{1}{2}\sqrt{8}\right).2\sqrt{6}\)

Answer 1

\(a,\sqrt{17-4\sqrt{9+4\sqrt{5}}}=\sqrt{17-4\sqrt{5+4\sqrt{5}+4}}=\sqrt{17-4\sqrt{\left(\sqrt{5}\right)^2+2.2\sqrt{5}+2^2}}=\sqrt{17-4\sqrt{\sqrt{\left(\sqrt{5}+2\right)^2}}}=\sqrt{17-4\sqrt{\sqrt{5}+2}}\) \(b,\sqrt{a};đk:a\ge0;2-3=-1< 0\Rightarrow sai\)

\(c,\sqrt{\left(\sqrt{3-3}\right)^2}.\sqrt{\frac{1}{3-\sqrt{3}}}=\sqrt{0^2}.\sqrt{\frac{1}{3-\sqrt{3}}}=0.\sqrt{\frac{1}{3-\sqrt{3}}}=0\)

\(d,\left(\sqrt{6}-3\sqrt{3}+5\sqrt{2}-\frac{1}{2}\sqrt{8}\right)2\sqrt{6}=\left(\sqrt{2}.\sqrt{3}-3\sqrt{3}+5\sqrt{2}-\sqrt{2}\right)2\sqrt{6}=\left[\sqrt{3}\left(\sqrt{2}-3\right)+\sqrt{2}.4\right]2\sqrt{6}=\left[2.\sqrt{3}.\sqrt{2}.\sqrt{3}\left(\sqrt{2}-3\right)+\sqrt{2}.\sqrt{2}.\sqrt{3}.2.4\right]=6\sqrt{2}\left(\sqrt{2}-3\right)+16\sqrt{3}\)