Đáp án:
\(a)\sqrt{27}-2\sqrt{48}-6\sqrt{\dfrac{1}{12}}\)
\(=\sqrt{9.3}-2\sqrt{16.3}-\sqrt{36.\dfrac{1}{12}}\)
\(=3\sqrt{3}-8\sqrt{3}-\sqrt{3}\)
\(=-6\sqrt{3}\)
\(b)\sqrt{(2\sqrt{3}-3\sqrt{2})^2}-\sqrt{19-3\sqrt{8}}+(\sqrt{2\sqrt{3}})^2\)
\(=|2\sqrt{3}-3\sqrt{2}|-\sqrt{19-6\sqrt{2}}+2\sqrt{3}\)
\(=3\sqrt{2}-2\sqrt{3}-\sqrt{18-2.3\sqrt{2}+1}+2\sqrt{3}\)
\(=3\sqrt{2}-\sqrt{(3\sqrt{2}-1)^2}\)
\(=3\sqrt{2}-3\sqrt{2}+1=1\)
\(c)\left(\dfrac{\sqrt{5}+\sqrt{2}}{\sqrt{5}-\sqrt{2}}+\dfrac{2}{\sqrt{10}+2}\right):\dfrac{5}{5-3\sqrt{10}}\)
\(=\left(\dfrac{(\sqrt{5}+\sqrt{2})^2}{5-2}+\dfrac{2(\sqrt{10}-2)}{10-4}\right):\dfrac{5}{5-3\sqrt{10}}\)
\(=\left(\dfrac{7+2\sqrt{10}}{3}+\dfrac{\sqrt{10}-2}{3}\right):\dfrac{5}{5-3\sqrt{10}}\)
\(=\left(\dfrac{7+2\sqrt{10}+\sqrt{10}-2}{3}\right).\dfrac{5-3\sqrt{10}}{5}\)
\(=\dfrac{(5+3\sqrt{10})(5-3\sqrt{10})}{3}\)
\(=\dfrac{25-90}{3}\)
\(=\dfrac{5-75}{3}=-25\)
\(d)\left(\dfrac{\sqrt{x}-1}{x-1}+\dfrac{x+\sqrt{x}+1}{x\sqrt{x}-1}\right)(x-1)(x \ge 0,x \ne 1)\)
\(=\left(\dfrac{\sqrt{x}-1}{(\sqrt{x}+1)(\sqrt{x}-1)}+\dfrac{x+\sqrt{x}+1}{(\sqrt{x}-1)(x+\sqrt{x}+1)}\right).(x-1)\)
\(=\left(\dfrac{1}{\sqrt{x}+1}+\dfrac{1}{\sqrt{x}-1}\right).(x-1)\)
\(=\left(\dfrac{\sqrt{x}-1+\sqrt{x}+1}{x-1}\right).(x-1)\)
\(=\dfrac{2\sqrt{x}}{x-1}.(x-1)\)
\(=2\sqrt{x}\)
