$\displaystyle \begin{array}{{>{\displaystyle}l}} a) \ 3\sqrt{18} -\sqrt{32} +4\sqrt{2} +\sqrt{162}\\ =3.3\sqrt{2} -4\sqrt{2} +4\sqrt{2} +9\sqrt{2}\\ =18\sqrt{2} \ \\ b) \ \sqrt{\left( 2-\sqrt{3}\right)^{2}} +\sqrt{\left( 2+\sqrt{3}\right)^{2}}\\ =|2-\sqrt{3} |+|2+\sqrt{3} |\\ =2-\sqrt{3} +2+\sqrt{3}\\ =4\\ c) \ \frac{a\sqrt{b} +b\sqrt{a}}{\sqrt{ab}} :\frac{1}{\sqrt{a} -\sqrt{b}}\\ =\frac{\sqrt{ab}\left(\sqrt{a} +\sqrt{b}\right)}{\sqrt{ab}} .\left(\sqrt{a} -\sqrt{b}\right)\\ =\left(\sqrt{a} +\sqrt{b}\right)\left(\sqrt{a} -\sqrt{b}\right)\\ =a-b\ \\ d) \ \sqrt{14+6\sqrt{5}} +\sqrt{\left( 3-\sqrt{5}\right)^{2}}\\ =\sqrt{9+2.3\sqrt{5} +5} +|3-\sqrt{5} |\\ =\sqrt{\left( 3+\sqrt{5}\right)^{2}} +3-\sqrt{5}\\ =3+\sqrt{5} +3-\sqrt{5}\\ =6\\ Câu\ 2:\ \\ a) \ \sqrt{4x^{2} -12x+9} =x-1\\ \sqrt{( 2x-3)^{2}} =1\\ \rightarrow |2x-3|=1\ \\ \rightarrow \begin{cases} 2x-3=1 & \\ 2x-3=-1 & \end{cases}\rightarrow \begin{cases} x=2 & \\ x=1 & \end{cases}\\ Tập\ nghiệm\ S=\{2;1\}\\ b)\sqrt{16x+16} -\sqrt{9x+9} +\sqrt{4x+4} +\sqrt{x+1} =16\\ 4\sqrt{x+1} -3\sqrt{x+1} +2\sqrt{x+1} +\sqrt{x+1} =16\\ 4\sqrt{x+1} =16\\ \rightarrow \sqrt{x+1} =4\ \\ DKXD\ :\ x+1\geqslant 0\ \\ \rightarrow x\geqslant -1\ \\ \rightarrow x+1=16\\ \rightarrow x=15\ ( tm) \ \\ Tập\ nghiệm\ S=\{15\}\\ A=\frac{-1}{x-\sqrt{x-2}} \ với\ x\geqslant 2\\ A=\frac{1}{\sqrt{x-2} +x} \ x\geqslant 2\ \\ Ta\ có\ :\sqrt{x-2} \geqslant 0\ với\ mọi\ x\ \\ \rightarrow Vì\ x\ \geqslant 2\ nên\ :\\ x+\sqrt{x-2} \geqslant 2\\ Do\ đó\ :\ \frac{1}{x+\sqrt{x-2}} \leqslant \frac{1}{2} \ \\ Dấu\ bằng\ xảy\ ra\ khi\ x=2\ \\ Vậy\ maxA=\frac{1}{2} \ tại\ x=2\ \end{array}$
