Đáp án:
$a)\dfrac{4\sqrt{3}}{3}\\ b)\dfrac{17\sqrt{6}}{6}\\ c)-\dfrac{3}{5}\\ d)2+\sqrt{3}$
Giải thích các bước giải:
$a)2\sqrt{\dfrac{3}{4}}-4\sqrt{\dfrac{1}{3}}+\dfrac{2}{3}\sqrt{\dfrac{75}{4}}\\ =2\dfrac{\sqrt{3}}{2}-4\dfrac{1}{\sqrt{3}}+\dfrac{2}{3}\sqrt{\dfrac{25.3}{4}}\\ =\sqrt{3}-\dfrac{4}{\sqrt{3}}+\dfrac{2}{3}.\dfrac{5}{2}.\sqrt{3}\\ =\sqrt{3}-\dfrac{4}{\sqrt{3}}+\dfrac{5\sqrt{3}}{3}\\ =\dfrac{3\sqrt{3}}{3}-\dfrac{4\sqrt{3}}{3}+\dfrac{5\sqrt{3}}{3}\\ =\dfrac{3\sqrt{3}-4\sqrt{3}+5\sqrt{3}}{3}\\ =\dfrac{4\sqrt{3}}{3}\\ b)\dfrac{2}{\sqrt{6}-2}+\dfrac{2}{\sqrt{6}+2}+\dfrac{5}{\sqrt{6}}\\ =\dfrac{2(\sqrt{6}+2)}{(\sqrt{6}-2)(\sqrt{6}+2)}+\dfrac{2(\sqrt{6}-2)}{(\sqrt{6}+2)(\sqrt{6}-2)}+\dfrac{5}{\sqrt{6}}\\ =\dfrac{2\sqrt{6}+4}{2}+\dfrac{2\sqrt{6}-4}{2}+\dfrac{5}{\sqrt{6}}\\ =\sqrt{6}+2+\sqrt{6}-2+\dfrac{5}{\sqrt{6}}\\ =2\sqrt{6}+\dfrac{5}{\sqrt{6}}\\ =\dfrac{2.6}{\sqrt{6}}+\dfrac{5}{\sqrt{6}}\\ =\dfrac{17}{\sqrt{6}}\\ =\dfrac{17\sqrt{6}}{6}\\ c)\left(2\sqrt{5}-\dfrac{3}{\sqrt{5}}-\dfrac{1}{2}\sqrt{80}\right):\sqrt{5}\\ =2-\dfrac{3}{5}-\dfrac{1}{2}\sqrt{16}\\ =2-\dfrac{3}{5}-\dfrac{1}{2}.4\\ =2-\dfrac{3}{5}-2\\ =-\dfrac{3}{5}\\ d)\sqrt{9+4\sqrt{5}}-\sqrt{8-2\sqrt{15}}\\ =\sqrt{5+2.\sqrt{5}.2+4}-\sqrt{5-2\sqrt{5}.\sqrt{3}+3}\\ =\sqrt{(\sqrt{5}+2)^2}-\sqrt{(\sqrt{5}-\sqrt{3})^2}\\ =\sqrt{5}+2-(\sqrt{5}-\sqrt{3})\\ =\sqrt{5}+2-\sqrt{5}+\sqrt{3}\\ =2+\sqrt{3}$
