Đáp án:

$\displaystyle \begin{array}{{>{\displaystyle}l}} 29.\ 2\sqrt{2} -\sqrt{6}\\ 30.\ 3-2\sqrt{2}\\ 31.\ \sqrt{24} +1\\ 32.\ \sqrt{18} -\sqrt{3}\\ 33.\ 2\sqrt{2} +\sqrt{6}\\ 34.\ \ 3+2\sqrt{2}\\ 35.\ \sqrt{5} +2\sqrt{2}\\ 36.\ 5-2\sqrt{2}\\ 37.\ 3\sqrt{2} +\sqrt{3}\\ 38.\ \sqrt{30} -\sqrt{6}\\ 39.\ 5-2\sqrt{2}\\ 40.\ \sqrt{45} +1\\ 41.\ 2\sqrt{5} -3\\ 42.\sqrt{30} +\sqrt{6}\\ 43.\ 3\sqrt{3} -2\sqrt{2}\\ 44.\ 5+2\sqrt{2}\\ 45.\ \sqrt{30} -\sqrt{6}\\ 46.\sqrt{45} -1\\ 47.\ 2\sqrt{5} +3\\ 48.\sqrt{3} +2\sqrt{6}\\ 49.5-2\sqrt{6}\\ 50.4\sqrt{6} -\sqrt{2}\\ 51.\ \frac{\sqrt{6}}{2} -\frac{\sqrt{2}}{2}\\ 52.\ \sqrt{5} +\sqrt{3} \end{array}$ 

Giải thích các bước giải:

$\displaystyle \begin{array}{{>{\displaystyle}l}} 29.\ \sqrt{14-8\sqrt{3}} =\sqrt{2}\sqrt{( 7-4\sqrt{3}} =\sqrt{2}\sqrt{( 7-4\sqrt{3}}\\ =\sqrt{2}\sqrt{\left( 2-\sqrt{3}\right)^{2}} =\sqrt{2}\left( 2-\sqrt{3}\right) =2\sqrt{2} -\sqrt{6}\\ 30.\ \sqrt{17-12\sqrt{2}} =\sqrt{\left( 3-2\sqrt{2}\right)^{2}} =3-2\sqrt{2}\\ 31.\ \sqrt{25+4\sqrt{6}} =\sqrt{25+2\sqrt{24}} =\sqrt{\left(\sqrt{24} +1\right)^{2}} =\sqrt{24} +1\\ 32.\ \sqrt{21-6\sqrt{6}} =\sqrt{21-2\sqrt{3}\sqrt{18}} =\sqrt{\left(\sqrt{3} -\sqrt{18}\right)^{2}} =\sqrt{18} -\sqrt{3}\\ 33.\ \sqrt{14+8\sqrt{3}} =\sqrt{2}\sqrt{( 7+4\sqrt{3}} =\sqrt{2}\sqrt{( 7+4\sqrt{3}}\\ =\sqrt{2}\sqrt{\left( 2+\sqrt{3}\right)^{2}} =\sqrt{2}\left( 2+\sqrt{3}\right) =2\sqrt{2} +\sqrt{6}\\ 34.\ \ \sqrt{17+12\sqrt{2}} =\sqrt{\left( 3+2\sqrt{2}\right)^{2}} =3+2\sqrt{2}\\ 35.\ \sqrt{13+4\sqrt{10}} =\sqrt{\left(\sqrt{5} +2\sqrt{2}\right)^{2}} =\sqrt{5} +2\sqrt{2}\\ 36.\ \sqrt{33-20\sqrt{2}} =\sqrt{\left( 5-2\sqrt{2}\right)^{2}} =5-2\sqrt{2}\\ 37.\ \sqrt{21+6\sqrt{6}} =\sqrt{\left( 3\sqrt{2} +\sqrt{3}\right)^{2}} =3\sqrt{2} +\sqrt{3}\\ 38.\ \sqrt{36-12\sqrt{5}} =\sqrt{3}\sqrt{12-4\sqrt{5}} =\sqrt{3}\sqrt{\left(\sqrt{10} -\sqrt{2}\right)^{2}} =\sqrt{30} -\sqrt{6}\\ 39.\ \sqrt{13-4\sqrt{10}} =\sqrt{\left( 2\sqrt{2} -5\right)^{2}} =5-2\sqrt{2}\\ 40.\ \sqrt{46+6\sqrt{5}} =\sqrt{\left(\sqrt{45} +1\right)^{2}} =\sqrt{45} +1\\ 41.\ \sqrt{29-12\sqrt{5}} =\sqrt{\left( 3-2\sqrt{5}\right)^{2}} =2\sqrt{5} -3\\ 42.\ \sqrt{36+12\sqrt{5}} =\sqrt{3}\sqrt{12+4\sqrt{5}} =\sqrt{3}\sqrt{\left(\sqrt{10} +\sqrt{2}\right)^{2}} =\sqrt{30} +\sqrt{6}\\ 43.\ \sqrt{35-12\sqrt{6}} =\sqrt{\left( 2\sqrt{2} -3\sqrt{3}\right)^{2}} =3\sqrt{3} -2\sqrt{2}\\ 44.\ \sqrt{33+20\sqrt{2}} =\sqrt{\left( 5+2\sqrt{2}\right)^{2}} =5+2\sqrt{2}\\ 45.\ \sqrt{36-12\sqrt{5}} =\sqrt{3}\sqrt{12-4\sqrt{5}} =\sqrt{3}\sqrt{\left(\sqrt{10} -\sqrt{2}\right)^{2}} =\sqrt{30} -\sqrt{6}\\ 46.\sqrt{46-6\sqrt{5}} =\sqrt{\left(\sqrt{45} -1\right)^{2}} =\sqrt{45} -1\\ 47.\ \sqrt{29+12\sqrt{5}} =\sqrt{\left( 2\sqrt{5} +3\right)^{2}} =2\sqrt{5} +3\\ 48.\ \sqrt{27+12\sqrt{2}} =\sqrt{3}\sqrt{9+4\sqrt{2}} =\sqrt{3}\sqrt{\left( 1+2\sqrt{2}\right)^{2}} =\sqrt{3} +2\sqrt{6}\\ 49.\sqrt{49-20\sqrt{6}} =\sqrt{\left( 2\sqrt{6} -5\right)^{2}} =5-2\sqrt{6}\\ 50.\sqrt{98-16\sqrt{3}} =\sqrt{2}\sqrt{49-8\sqrt{3}} =\sqrt{2}\sqrt{\left(\sqrt{48} -1\right)^{2}}\\ =\sqrt{2}\left(\sqrt{48} -1\right) =4\sqrt{6} -\sqrt{2}\\ 51.\ \sqrt{2-\sqrt{3}} =\sqrt{2}\sqrt{1-\frac{\sqrt{3}}{2}} =\ \sqrt{2}\sqrt{\left(\frac{1}{2} -\frac{\sqrt{3}}{2}\right)^{2}} =\ \sqrt{2}\left(\frac{\sqrt{3}}{2} -\frac{1}{2}\right) =\frac{\sqrt{6}}{2} -\frac{\sqrt{2}}{2}\\ 52.\ \sqrt{4+\sqrt{15}} =\sqrt{2}\sqrt{2+\frac{\sqrt{15}}{2}} =2\sqrt{1+\frac{\sqrt{15}}{4}} =2\left(\frac{\sqrt{5}}{2} +\frac{\sqrt{3}}{2}\right) =\sqrt{5} +\sqrt{3} \end{array}$