Đáp án:
$1)\left\{\begin{array}{cc} \sqrt{x-2}-1,& x\ge 3\\ 1-\sqrt{x-2},&2<x<3\end{array} \right.\\ 2)\sqrt{x}+1\\ 3)\sqrt{x-3}+1\\ 4)\left\{\begin{array}{cc} \sqrt{x+1}-1,& x\ge 0\\ 1-\sqrt{x+1},&-1<x<0\end{array} \right.\\ 5) \left\{\begin{array}{cc} \sqrt{2(x-1)}-1,& x \ge \dfrac{3}{2}\\ 1-\sqrt{2(x-1)},&1<x<\dfrac{3}{2} \end{array} \right.\\ 6)\sqrt{2x}+1\\ 7) \sqrt{x-2}-\sqrt{x-3}\\ 8)\sqrt{x+2}+\sqrt{x-3}$
Giải thích các bước giải:
$1)\sqrt{x-1-2\sqrt{x-2}}\\ =\sqrt{x-2-2\sqrt{x-2}+1}\\ =\sqrt{(\sqrt{x-2}-1)^2}\\ =|\sqrt{x-2}-1|\\ = \left\{\begin{array}{cc} \sqrt{x-2}-1,& \sqrt{x-2}\ge 1\\ 1-\sqrt{x-2},&\sqrt{x-2}<1\end{array} \right.\\ =\left\{\begin{array}{cc} \sqrt{x-2}-1,& x\ge 3\\ 1-\sqrt{x-2},&2<x<3\end{array} \right.\\ 2)\sqrt{x+1+2\sqrt{x}}\\ =\sqrt{x+2\sqrt{x}+1}\\ =\sqrt{(\sqrt{x}+1)^2}\\ =|\sqrt{x}+1|\\ =\sqrt{x}+1\\ 3)\sqrt{x-2+2\sqrt{x-3}}\\ =\sqrt{x-3+2\sqrt{x-3}+1}\\ =\sqrt{(\sqrt{x-3}+1)^2}\\ =|\sqrt{x-3}+1|\\ =\sqrt{x-3}+1\\ 4)\sqrt{x+2-2\sqrt{x+1}}\\ =\sqrt{x+1-2\sqrt{x+1}+1}\\ =\sqrt{(\sqrt{x+1}-1)^2}\\ =|\sqrt{x+1}-1|\\ = \left\{\begin{array}{cc} \sqrt{x+1}-1,& \sqrt{x+1}\ge 1\\ 1-\sqrt{x+1},&\sqrt{x+1}<1\end{array} \right.\\ =\left\{\begin{array}{cc} \sqrt{x+1}-1,& x\ge 0\\ 1-\sqrt{x+1},&-1<x<0\end{array} \right.\\ 5)\sqrt{2x-1-2\sqrt{2(x-1)}}\\ =\sqrt{2(x-1)-2\sqrt{2(x-1)}+1}\\ =\sqrt{(\sqrt{2(x-1)}-1)^2}\\ =|\sqrt{2(x-1)}-1|\\ = \left\{\begin{array}{cc} \sqrt{2(x-1)}-1,& \sqrt{2(x-1)} \ge 1\\ 1-\sqrt{2(x-1)},&\sqrt{2(x-1)}<1 \end{array} \right.\\ = \left\{\begin{array}{cc} \sqrt{2(x-1)}-1,& 2(x-1) \ge 1\\ 1-\sqrt{2(x-1)},&0<2(x-1)<1 \end{array} \right.\\ = \left\{\begin{array}{cc} \sqrt{2(x-1)}-1,& x \ge \dfrac{3}{2}\\ 1-\sqrt{2(x-1)},&1<x<\dfrac{3}{2} \end{array} \right.\\ 6)\sqrt{2x+1+2\sqrt{2x}}\\ =\sqrt{2x+2\sqrt{2x}+1}\\ =\sqrt{(\sqrt{2x}+1)^2}\\ =|\sqrt{2x}+1|\\ =\sqrt{2x}+1\\ 7)\sqrt{2x-5-2\sqrt{(x-2)(x-3)}}\\ =\sqrt{x-2-2\sqrt{(x-2)(x-3)+x-3}}\\ =\sqrt{(\sqrt{x-2}-\sqrt{x-3})^2}\\ =|\sqrt{x-2}-\sqrt{x-3}|\\ = \sqrt{x-2}-\sqrt{x-3}\\ 8)\sqrt{2x-1+2\sqrt{(x+2)(x-3)}}\\ =\sqrt{x+2+2\sqrt{(x+2)(x-3)+x-3}}\\ =\sqrt{(\sqrt{x+2}+\sqrt{x-3})^2}\\ =|\sqrt{x+2}+\sqrt{x-3}|\\ = \sqrt{x+2}+\sqrt{x-3}$
