Bài 1:
a: \(\sqrt{48}+\sqrt{27}-\sqrt{75}+\dfrac{2}{3}\cdot\sqrt{\dfrac{1}{3}}\)
\(=4\sqrt{3}+3\sqrt{3}-5\sqrt{3}+\dfrac{2}{3}\cdot\dfrac{1}{\sqrt{3}}\)
\(=2\sqrt{3}+\dfrac{2}{3\sqrt{3}}\)
\(=2\sqrt{3}+\dfrac{2\sqrt{3}}{9}\)
\(=\dfrac{20\sqrt{3}}{9}\)
b: \(\left(\dfrac{1}{2}\sqrt{\dfrac{1}{2}}-\dfrac{3}{2}\cdot\sqrt{2}+\dfrac{4}{5}\cdot\sqrt{200}\right):\dfrac{1}{8}\)
\(=\left(\dfrac{1}{\sqrt{2}\cdot2}-\dfrac{3}{2}\cdot\sqrt{2}+\dfrac{4}{5}\cdot10\sqrt{2}\right):\dfrac{1}{8}\)
\(=\left(\dfrac{1}{2\sqrt{2}}-\dfrac{3\sqrt{2}}{2}+8\sqrt{2}\right)\cdot8\)
\(=\dfrac{8}{2\sqrt{2}}-8\cdot\dfrac{3\sqrt{2}}{2}+8\cdot8\sqrt{2}\)
\(=2\sqrt{2}-12\sqrt{2}+64\sqrt{2}=54\sqrt{2}\)
c: \(0,2\cdot\sqrt{\left(-5\right)^4\cdot2}+\sqrt{2\cdot\left(-5\right)^2}-5\cdot\sqrt{3-2\sqrt{2}}\)
\(=0,2\cdot5^2\cdot\sqrt{2}+5\sqrt{2}-5\cdot\sqrt{\left(\sqrt{2}-1\right)^2}\)
\(=5\sqrt{2}+5\sqrt{2}-5\left(\sqrt{2}-1\right)\)
\(=10\sqrt{2}-5\sqrt{2}+5=5\sqrt{2}+5\)
d: \(\dfrac{3}{\sqrt{5}-2}+\dfrac{2}{\sqrt{5}+3}-\dfrac{1}{4+\sqrt{5}}\)
\(=\dfrac{3\left(\sqrt{5}+2\right)}{5-4}+\dfrac{2\left(3-\sqrt{5}\right)}{9-5}-\dfrac{4-\sqrt{5}}{16-5}\)
\(=3\sqrt{5}+6+\dfrac{3-\sqrt{5}}{2}-\dfrac{4-\sqrt{5}}{11}\)
\(=3\sqrt{5}+6+\dfrac{3}{2}-\dfrac{1}{2}\sqrt{5}-\dfrac{4}{11}+\dfrac{\sqrt{5}}{11}\)
\(=\dfrac{57}{22}\sqrt{5}+\dfrac{157}{22}\)
e: \(\sqrt{\left(2-\sqrt{3}\right)^2}+\sqrt{4-2\sqrt{3}}\)
\(=\left|2-\sqrt{3}\right|+\sqrt{\left(\sqrt{3}-1\right)^2}\)
\(=2-\sqrt{3}+\left|\sqrt{3}-1\right|\)
\(=2-\sqrt{3}+\sqrt{3}-1=2-1=1\)
f: \(\sqrt{15-6\sqrt{6}}+\sqrt{33-12\sqrt{6}}\)
\(=\sqrt{15-2\cdot3\sqrt{6}}+\sqrt{33-2\cdot6\sqrt{6}}\)
\(=\sqrt{\left(3-\sqrt{6}\right)^2}+\sqrt{24-2\cdot2\sqrt{6}\cdot3+9}\)
\(=\sqrt{\left(3-\sqrt{6}\right)^2}+\sqrt{\left(2\sqrt{6}-3\right)^2}\)
\(=\left|3-\sqrt{6}\right|+\left|2\sqrt{6}-3\right|\)
\(=3-\sqrt{6}+2\sqrt{6}-3=\sqrt{6}\)
