$\displaystyle \begin{array}{{>{\displaystyle}l}} Bài\ 3:\\ 1.\ \ \frac{x+\sqrt{x}}{1+\sqrt{x}} \ xác\ định\ \Leftrightarrow x\geqslant 0\\ \frac{x+\sqrt{x}}{1+\sqrt{x}} =\frac{\sqrt{x}\left(\sqrt{x} +1\right)}{1+\sqrt{x}} =\sqrt{x}\\ 2.\ \frac{a-2\sqrt{a}}{2-\sqrt{a}} \ xác\ định\ \Leftrightarrow a\geqslant 0\ và\ a\neq 4\\ \frac{a-2\sqrt{a}}{2-\sqrt{a}} =\frac{\sqrt{a}\left(\sqrt{a} -2\right)}{2-\sqrt{a}} =-\sqrt{a}\\ 3.\ \frac{1-a}{1+\sqrt{a}} \ xác\ định\ \Leftrightarrow a\geqslant 0\\ \ \frac{1-a}{1+\sqrt{a}} =\frac{\left( 1+\sqrt{a}\right)\left( 1-\sqrt{a}\right)}{1+\sqrt{a}} =1-\sqrt{a}\\ 4.\ \frac{x^{2} -3}{x+\sqrt{3\ }} \ xác\ định\ \Leftrightarrow x\in R\\ \frac{x^{2} -3}{x+\sqrt{3\ }} =\frac{\left( x+\sqrt{3\ }\right)\left( x-\sqrt{3\ }\right)}{x+\sqrt{3\ }} =x-\sqrt{3}\\ 5.\ \frac{4-a}{\sqrt{a} -2} \ xác\ định\ \Leftrightarrow a\geqslant 0\ và\ a\neq 4\\ \frac{4-a}{\sqrt{a} -2} =\frac{-\left(\sqrt{a} -2\right)\left(\sqrt{a} +2\right)}{\sqrt{a} -2} =-\sqrt{a} -2\\ 6.\ \frac{a-1}{\sqrt{a} -1} \ xác\ định\ \Leftrightarrow a\geqslant 0;\ a\neq 1\\ \frac{a-1}{\sqrt{a} -1} =\frac{\left(\sqrt{a} -1\right)\left(\sqrt{a} +1\right)}{\sqrt{a} -1} =\sqrt{a} +1\\ 7.\ \frac{a-b}{\sqrt{a} +\sqrt{b}} \ xác\ định\ \Leftrightarrow a\geqslant 0;b\geqslant 0\ ;\ a\ và\ b\ không\ đồng\ thời\ bằng\ 0\\ \frac{a-b}{\sqrt{a} +\sqrt{b}} =\frac{\left(\sqrt{a} +\sqrt{b}\right)\left(\sqrt{a} -\sqrt{b}\right)}{\sqrt{a} +\sqrt{b}} =\sqrt{a} -\sqrt{b}\\ 8.\ \frac{a-b}{\sqrt{a} -\sqrt{b}} \ xác\ định\ \Leftrightarrow a\geqslant 0;b\geqslant 0\ ;\ a\neq b\\ \frac{a-b}{\sqrt{a} -\sqrt{b}} =\frac{\left(\sqrt{a} +\sqrt{b}\right)\left(\sqrt{a} -\sqrt{b}\right)}{\sqrt{a} -\sqrt{b}} =\sqrt{a} +\sqrt{b}\\ 9.\ \frac{a+4\sqrt{a} +4}{\sqrt{a} +2} \ xác\ định\ \Leftrightarrow a\geqslant 0\\ \frac{a+4\sqrt{a} +4}{\sqrt{a} +2} =\frac{\left(\sqrt{a} +2\right)^{2}}{\sqrt{a} +2} =\sqrt{a} +2\\ 10.\ \frac{a+b-2\sqrt{ab}}{\sqrt{a} -\sqrt{b}} xác\ định\ \Leftrightarrow a\geqslant 0;b\geqslant 0\ ;\ a\neq b\\ \frac{a+b-2\sqrt{ab}}{\sqrt{a} -\sqrt{b}} =\frac{\left(\sqrt{a} -\sqrt{b}\right)^{2}}{\sqrt{a} -\sqrt{b}} =\sqrt{a} -\sqrt{b}\\ 11.\ \frac{1+a\sqrt{a}}{1+\sqrt{a}} xác\ định\ \Leftrightarrow a\geqslant 0\\ \frac{1+a\sqrt{a}}{1+\sqrt{a}} =\frac{\left( 1+\sqrt{a}\right)\left( 1-\sqrt{a} +a\right)}{1+\sqrt{a}} =1-\sqrt{a} +a\\ 12.\ \frac{a\sqrt{a} -1}{\sqrt{a} -1} xác\ định\ \Leftrightarrow a\geqslant 0;a\neq 1\\ \frac{a\sqrt{a} -1}{\sqrt{a} -1} =\frac{\left(\sqrt{a} -1\right)\left( 1+\sqrt{a} +a\right)}{\sqrt{a} -1} =1+\sqrt{a} +a\\ 13.\ \frac{a\sqrt{a} +b\sqrt{b}}{\sqrt{a} +\sqrt{b}} \ xác\ định\ \Leftrightarrow a\geqslant 0;b\geqslant 0\ ;\ a\ và\ b\ không\ đồng\ thời\ bằng\ 0\\ \frac{a\sqrt{a} +b\sqrt{b}}{\sqrt{a} +\sqrt{b}} =\frac{\left(\sqrt{a} +\sqrt{b}\right)\left( a-\sqrt{ab} +b\right)}{\sqrt{a} +\sqrt{b}} =a-\sqrt{ab} +b\\ 14.\ \ \frac{a\sqrt{a} -b\sqrt{b}}{\sqrt{a} -\sqrt{b}} \ xác\ định\ \Leftrightarrow a\geqslant 0;b\geqslant 0\ ;\ a\neq b\\ \frac{a\sqrt{a} -b\sqrt{b}}{\sqrt{a} -\sqrt{b}} =\frac{\left(\sqrt{a} -\sqrt{b}\right)\left( a+\sqrt{ab} +b\right)}{\sqrt{a} -\sqrt{b}} =a+\sqrt{ab} +b\\ 15.\ \frac{\sqrt{x} +\sqrt{y}}{\sqrt{x} -\sqrt{y}} \ xác\ định\ \Leftrightarrow x\geqslant 0;\ y\geqslant 0;\ x\neq y\\ \frac{\sqrt{x} +\sqrt{y}}{\sqrt{x} -\sqrt{y}} =\frac{\left(\sqrt{x} +\sqrt{y}\right)^{2}}{\left(\sqrt{x} -\sqrt{y}\right)\left(\sqrt{x} +\sqrt{y}\right)} =\frac{x+y+2\sqrt{xy}}{x-y}\\ 16.\ \frac{1+\sqrt{a}}{1-\sqrt{a} \ } \ xác\ định\ \Leftrightarrow a\geqslant 0;\ a\neq 1\\ \frac{1+\sqrt{a}}{1-\sqrt{a} \ } =\frac{\left( 1+\sqrt{a}\right)^{2}}{\left( 1-\sqrt{a} \ \right)\left( 1+\sqrt{a}\right)} =\frac{1+a+2\sqrt{a}}{1-a \ }\\ Bài\ 4:\\ 1.\ \frac{1}{\sqrt{3} +\sqrt{2} +1} =\frac{\sqrt{3} -\sqrt{2} -1}{\left(\sqrt{3} +\sqrt{2} +1\right)\left(\sqrt{3} -\sqrt{2} -1\right)}\\ =\frac{\sqrt{3} -\sqrt{2} -1}{\left(\sqrt{3}\right)^{2} -\left(\sqrt{2} +1\right)^{2}} =\frac{\sqrt{3} -\sqrt{2} -1}{3-2-2\sqrt{2} -1}\\ =\frac{\sqrt{3} -\sqrt{2} -1}{-2\sqrt{2}} =\frac{-\sqrt{2}\left(\sqrt{3} -\sqrt{2} -1\right)}{4}\\ 2.\ \frac{2\sqrt{3}}{\sqrt{2} +\sqrt{3} +\sqrt{5}} =\frac{2\sqrt{3}\left(\sqrt{2} +\sqrt{3} -\sqrt{5}\right)}{\left(\sqrt{2} +\sqrt{3} +\sqrt{5}\right)\left(\sqrt{2} +\sqrt{3} -\sqrt{5}\right)}\\ =\frac{2\sqrt{3}\left(\sqrt{2} +\sqrt{3} -\sqrt{5}\right)}{\left(\sqrt{2} +\sqrt{3}\right)^{2} -\left(\sqrt{5}\right)^{2}} =\frac{2\sqrt{3}\left(\sqrt{2} +\sqrt{3} -\sqrt{5}\right)}{5+2\sqrt{6} -5}\\ =\frac{2\sqrt{3}\left(\sqrt{2} +\sqrt{3} -\sqrt{5}\right)}{2\sqrt{6}} =\frac{\sqrt{6}\sqrt{3}\left(\sqrt{2} +\sqrt{3} -\sqrt{5}\right)}{6}\\ 3.\ \frac{\sqrt{6}}{3+\sqrt{2} -\sqrt{3}} =\frac{6\left( 3-\sqrt{2} +\sqrt{3}\right)}{\left( 3+\sqrt{2} -\sqrt{3}\right)\left( 3-\sqrt{2} +\sqrt{3}\right)}\\ =\frac{6\left( 3-\sqrt{2} +\sqrt{3}\right)}{3^{2} -\left(\sqrt{2} -\sqrt{3}\right)^{2}} =\frac{6\left( 3-\sqrt{2} +\sqrt{3}\right)}{9-\left( 5-2\sqrt{6}\right)}\\ =\frac{6\left( 3-\sqrt{2} +\sqrt{3}\right)}{4+2\sqrt{6}} =\frac{3\left( 3-\sqrt{2} +\sqrt{3}\right)}{2+\sqrt{6}}\\ =\frac{3\left( 3-\sqrt{2} +\sqrt{3}\right)\left( 2+\sqrt{6}\right)}{4-6} =\frac{3\left( 3-\sqrt{2} +\sqrt{3}\right)\left( 2+\sqrt{6}\right)}{-2}\\ 4.\ \frac{2\sqrt{3}}{\sqrt{5} +\sqrt{6} +\sqrt{7}} =\frac{2\sqrt{3}\left(\sqrt{5} -\sqrt{6} -\sqrt{7}\right)}{\left(\sqrt{5} +\sqrt{6} +\sqrt{7}\right)\left(\sqrt{5} -\sqrt{6} -\sqrt{7}\right)}\\ =\frac{2\sqrt{3}\left(\sqrt{5} -\sqrt{6} -\sqrt{7}\right)}{\left(\sqrt{5}\right)^{2} -\left(\sqrt{6} +\sqrt{7}\right)^{2}} =\frac{2\sqrt{3}\left(\sqrt{5} -\sqrt{6} -\sqrt{7}\right)}{5-\left( 6+7+2\sqrt{42}\right)}\\ =\frac{2\sqrt{3}\left(\sqrt{5} -\sqrt{6} -\sqrt{7}\right)}{-8-2\sqrt{42}} =\frac{-\sqrt{3}\left(\sqrt{5} -\sqrt{6} -\sqrt{7}\right)}{4+\sqrt{42}}\\ =\frac{-\sqrt{3}\left(\sqrt{5} -\sqrt{6} -\sqrt{7}\right)\left( 4-\sqrt{42}\right)}{16-42}\\ =\frac{\sqrt{3}\left(\sqrt{5} -\sqrt{6} -\sqrt{7}\right)\left( 4-\sqrt{42}\right)}{26}\\ 5.\ \frac{1}{2+\sqrt{3} +\sqrt{6}} =\frac{2+\sqrt{3} +\sqrt{6}}{\left( 2+\sqrt{3} +\sqrt{6}\right)\left( 2+\sqrt{3} -\sqrt{6}\right)}\\ =\frac{2+\sqrt{3} +\sqrt{6}}{\left( 2+\sqrt{3}\right)^{2} -\left(\sqrt{6}\right)^{2}} =\frac{2+\sqrt{3} +\sqrt{6}}{6+2\sqrt{3} -6}\\ =\frac{2+\sqrt{3} +\sqrt{6}}{2\sqrt{3}} =\frac{\left( 2+\sqrt{3} +\sqrt{6}\right)\sqrt{3}}{6}\\ 6.\ \frac{1+3\sqrt{2} -2\sqrt{3}}{\sqrt{6} +\sqrt{3} +\sqrt{2}} =\frac{\left( 1+3\sqrt{2} -2\sqrt{3}\right)\left(\sqrt{3} +\sqrt{2} -\sqrt{6}\right)}{\left(\sqrt{6} +\sqrt{3} +\sqrt{2}\right)\left(\sqrt{3} +\sqrt{2} -\sqrt{6}\right)}\\ =\frac{\left( 1+3\sqrt{2} -2\sqrt{3}\right)\left(\sqrt{3} +\sqrt{2} -\sqrt{6}\right)}{\left(\sqrt{3} +\sqrt{2}\right)^{2} -\left(\sqrt{6}\right)^{2}}\\ =\frac{\left( 1+3\sqrt{2} -2\sqrt{3}\right)\left(\sqrt{3} +\sqrt{2} -\sqrt{6}\right)}{5+2\sqrt{6} -6}\\ =\frac{\left( 1+3\sqrt{2} -2\sqrt{3}\right)\left(\sqrt{3} +\sqrt{2} -\sqrt{6}\right)}{2\sqrt{6} -1}\\ =\frac{\left( 1+3\sqrt{2} -2\sqrt{3}\right)\left(\sqrt{3} +\sqrt{2} -\sqrt{6}\right)\left( 2\sqrt{6} +1\right)}{\left( 2\sqrt{6} -1\right)\left( 2\sqrt{6} +1\right)}\\ =\frac{\left( 1+3\sqrt{2} -2\sqrt{3}\right)\left(\sqrt{3} +\sqrt{2} -\sqrt{6}\right)\left( 2\sqrt{6} +1\right)}{23}\\ Bài\ 5:\\ 1.\ =\frac{\sqrt{6}\left(\sqrt{6} -1\right)}{\sqrt{6} -1} +\frac{\sqrt{6}\left(\sqrt{6} +1\right)}{\sqrt{6}}\\ =\sqrt{6} +\sqrt{6} +1\\ =2\sqrt{6} +1\\ 2.\ =\frac{6\left( 1-\sqrt{3}\right)}{1-\sqrt{3}} +\frac{3\left(\sqrt{3} +1\right)}{\sqrt{3} +1}\\ =6+3=9\\ 3.\ =\frac{\sqrt{3}\left(\sqrt{3} +1\right)}{\sqrt{3}} +\frac{\sqrt{3}\left(\sqrt{2} -1\right)}{1-\sqrt{2}}\\ =\sqrt{3} +1-\sqrt{3} =1\\ 4.\ \frac{\sqrt{2}\left(\sqrt{2} -1\right)}{1-\sqrt{2}} +\frac{\sqrt{2}\left( 1-\sqrt{3}\right)}{\sqrt{3} -1}\\ =-\sqrt{2} -\sqrt{2} =-2\sqrt{2}\\ 5.\ \frac{\sqrt{2}\left(\sqrt{5} -1\right)}{\sqrt{5} -1} +\frac{\sqrt{2}\left(\sqrt{2} -1\right)}{\sqrt{2} -1}\\ =\sqrt{2} +\sqrt{2} =2\sqrt{2}\\ 6.\ \frac{\sqrt{5}\left(\sqrt{3} -1\right)}{\sqrt{3} -1} +\frac{\sqrt{5}\left(\sqrt{5} -2\right)}{2\left(\sqrt{5} -2\right)}\\ =\sqrt{5} +\frac{\sqrt{5}}{2} =\frac{3\sqrt{5}}{2}\\ 7.\ \frac{\sqrt{3}\left(\sqrt{5} -2\right)}{\sqrt{5} -2} +\frac{\left(\sqrt{3} +\sqrt{2}\right) 2\sqrt{3}}{\sqrt{3} +\sqrt{2}}\\ =\sqrt{3} +2\sqrt{3} =3\sqrt{3}\\ 8.\ \frac{3\sqrt{2}\left( 1-\sqrt{2}\right)}{\sqrt{2} -1} +\frac{2\sqrt{2}\left( 3-\sqrt{2}\right)}{\sqrt{2} -3}\\ =-3\sqrt{2} -2\sqrt{2} \end{array}$