Đáp án:

 $\displaystyle \begin{array}{{>{\displaystyle}l}} Câu\ 90:\\ A==\frac{-\sqrt{x}}{\sqrt{x} +4}\\ Câu\ 92:\\ a.P==\frac{4\sqrt{x}}{x+4}\\ b.x=0\ hoặc\ x=4\ \\ Câu\ 93:\\ A=\frac{x+4}{2( x-4)}\\ Câu\ 94:\\ A=\frac{1}{a-1}\\ Câu\ 95:\\ A=\frac{\sqrt{x}}{\left( x+\sqrt{x} +1\right)}\\ Câu\ 96:\\ a.\ A=5\sqrt{2}\\ b.\ B=-\sqrt{a}\\ Câu\ 97:\\ P=2\\ Câu\ 98:\\ P=\frac{-2}{x+\sqrt{x} +1} \end{array}$

Giải thích các bước giải:

 $\displaystyle \begin{array}{{>{\displaystyle}l}} Bài\ 90:\\ A=\frac{2}{\sqrt{x} +4} +\frac{2}{\sqrt{x} -4} -\frac{x}{x-16}\\ ĐKXĐ:x\geqslant 0\ và\ x\neq 16\\ A=\frac{2.\left(\sqrt{x} -4\right) +2.\left(\sqrt{x} +4\right) -x}{\left(\sqrt{x} -4\right)\left(\sqrt{x} +4\right)} =\frac{-\sqrt{x}\left(\sqrt{x} -4\right)}{\left(\sqrt{x} -4\right)\left(\sqrt{x} +4\right)} =\frac{-\sqrt{x}}{\sqrt{x} +4}\\ Bài\ 92:\\ P=\left(\frac{\sqrt{x} +1}{\sqrt{x} -1} -\frac{\sqrt{x} -1}{\sqrt{x} +1} -\frac{8\sqrt{x}}{x-1}\right) :\left(\frac{\sqrt{x} -x-3}{x-1} -\frac{1}{\sqrt{x} -1}\right)\\ a.\ ĐKXĐ:x\geqslant 0,x\neq 1\\ P=\left(\frac{\left(\sqrt{x} +1\right)^{2} -\left(\sqrt{x} - 1\right)^{2} -8\sqrt{x}}{x-1}\right) :\left(\frac{\sqrt{x} -x-3-\sqrt{x} -1}{x-1}\right)\\ =\frac{-4\sqrt{x}}{-x-4} =\frac{4\sqrt{x}}{x+4}\\ b.\ Vì\ x\geqslant 0,x\neq 1\Rightarrow P=\frac{4\sqrt{x}}{x+4} \geqslant 0\\ Ta\ có:\ 1-P=1-\frac{4\sqrt{x}}{x+4} =\frac{x-4\sqrt{x} +4}{x+4} =\frac{\left(\sqrt{x} -2\right)^{2}}{x+4} \geqslant 0\Rightarrow P\leqslant 1\\ Do\ đó\ 0\leqslant P\leqslant 1\ mà\ P\ nguyên\ nên\ P=0\ hoặc\ P=1\\ Với\ P=0\ thì\ x=0( t/m)\\ Với\ P=1\ thì\ \sqrt{x} -2=0\Rightarrow x=4( t/m)\\ Vậy\ x=0\ hoặc\ x=4\ thì\ P\ nguyên\\ Bài\ 93:\\ A=\frac{8+x\left( 1+\sqrt{x-2\sqrt{x} +1}\right)}{( x-4)\left( x-2\sqrt{x} +4\right)} +\frac{x-3\sqrt{x}}{2\left( x-\sqrt{x} -6\right)}\\ ĐKXĐ:\ x >1;x\neq 4;x\neq 9\\ A=\frac{8+x\left( 1+\left(\sqrt{x} -1\right)\right)}{( x-4)\left( x-2\sqrt{x} +4\right)} +\frac{\sqrt{x}\left(\sqrt{x} -3\right)}{2\left(\sqrt{x} +2\right)\left(\sqrt{x} -3\right)}\\ =\frac{8+x\sqrt{x}}{( x-4)\left( x-2\sqrt{x} +4\right)} +\frac{\sqrt{x}}{2.\left(\sqrt{x} +2\right)}\\ =\frac{\left(\sqrt{x} +2\right)\left( x-2\sqrt{x} +4\right)}{\left(\sqrt{x} +2\right)\left(\sqrt{x} -2\right)\left( x-2\sqrt{x} +4\right)} +\frac{\sqrt{x}}{2.\left(\sqrt{x} +2\right)}\\ =\frac{1}{\sqrt{x} -2} +\frac{\sqrt{x}}{2\left(\sqrt{x} +2\right)} =\frac{2\left(\sqrt{x} +2\right) +\sqrt{x}\left(\sqrt{x} -2\right)}{2\left(\sqrt{x} +2\right)\left(\sqrt{x} -2\right)}\\ =\frac{x-2\sqrt{x} +2\sqrt{x} +4}{2( x-4)} =\frac{x+4}{2( x-4)}\\ Vậy\ với\ x >1;x\neq 4;x\neq 9\ thì\ A=\frac{x+4}{2( x-4)}\\ Bài\ 94:\\ A=\frac{1}{a^{2} -\sqrt{a}} :\frac{\sqrt{a} +1}{\sqrt{a} +a+a\sqrt{a}} ,\ a >0;\ a\neq 1\\ A=\frac{1}{\sqrt{a}\left( a\sqrt{a} -1\right)} :\frac{\sqrt{a} +1}{\sqrt{a}\left( a+\sqrt{a} +1\right)}\\ =\frac{1}{\sqrt{a}\left(\sqrt{a} -1\right)\left( a+\sqrt{a} +1\right)} :\frac{\sqrt{a} +1}{\sqrt{a}\left( a+\sqrt{a} +1\right)}\\ =\frac{\sqrt{a}\left( a+\sqrt{a} +1\right)}{\sqrt{a}\left(\sqrt{a} -1\right)\left( a+\sqrt{a} +1\right)\left(\sqrt{a} +1\right)} =\frac{1}{\left(\sqrt{a} -1\right)\left(\sqrt{a} +1\right)}\\ =\frac{1}{a-1}\\ Bài\ 95:\\ A=\frac{x+2}{x\sqrt{x} -1} +\frac{\sqrt{x} +1}{x+\sqrt{x} +1} -\frac{1}{\sqrt{x} -1} ,\ x\geqslant 0;x\neq 1\\ A=\frac{x+2}{\left(\sqrt{x} -1\right)\left( x+\sqrt{x} +1\right)} +\frac{\sqrt{x} +1}{x+\sqrt{x} +1} -\frac{1}{\sqrt{x} -1}\\ =\frac{x+2+\left(\sqrt{x} +1\right)\left(\sqrt{x} -1\right) -x-\sqrt{x} -1}{\left(\sqrt{x} -1\right)\left( x+\sqrt{x} +1\right)}\\ =\frac{x+2+x-1-x-\sqrt{x} -1}{\left(\sqrt{x} -1\right)\left( x+\sqrt{x} +1\right)} =\frac{x-\sqrt{x}}{\left(\sqrt{x} -1\right)\left( x+\sqrt{x} +1\right)}\\ =\frac{\sqrt{x}\left(\sqrt{x} -1\right)}{\left(\sqrt{x} -1\right)\left( x+\sqrt{x} +1\right)} =\frac{\sqrt{x}}{\left( x+\sqrt{x} +1\right)}\\ Bài\ 96:\\ a.\ A=2\sqrt{8} -5\sqrt{18} +4\sqrt{32}\\ =2.\sqrt{4.2} -5\sqrt{9.2} +4\sqrt{16.2} =4\sqrt{2} -15\sqrt{2} +16\sqrt{2} =5\sqrt{2}\\ b.\ B=\frac{a-\sqrt{a}}{a-2\sqrt{a} +1} .\left( 1-\sqrt{a}\right) \ với\ a >1\\ B=\frac{\sqrt{a}\left(\sqrt{a} -1\right)}{\left(\sqrt{a} -1\right)^{2}} .\left(\sqrt{a} -1\right) =-\sqrt{a}\\ Bài\ 97:\\ P=\sqrt{3+2\sqrt{2}} -\sqrt{2+\sqrt{2} -\sqrt{19-6\sqrt{2}}}\\ =\sqrt{3+2\sqrt{2}} -\sqrt{2+\sqrt{2} -\left( 3\sqrt{2} -1\right)}\\ =\sqrt{3+2\sqrt{2}} -\sqrt{3-2\sqrt{2}} =2\\ Bài\ 98:\\ \\ P=\left(\frac{x+2}{x\sqrt{x} -1} +\frac{\sqrt{x}}{x+\sqrt{x} +1} +\frac{1}{1-\sqrt{x}}\right) :\frac{\sqrt{x} -1}{2} \ với\ 0\leqslant x\neq 1\\ P=\frac{( x+2) +\sqrt{x}\left(\sqrt{x} -1\right) -\left( x+\sqrt{x} +1\right)}{\left( 1-\sqrt{x}\right) .\left( x+\sqrt{x} +1\right)} .\frac{2}{\sqrt{x} -1}\\ =\frac{\left(\sqrt{x} -1\right)^{2}}{\left( 1-\sqrt{x}\right) .\left( x+\sqrt{x} +1\right)} .\frac{2}{\sqrt{x} -1}\\ =\frac{-2}{x+\sqrt{x} +1}\\ \\ \end{array}$