- \(\frac{\sqrt{27\left(1-\sqrt{3}\right)^4}}{3\sqrt{15}}=\frac{\sqrt{3.3^2\left(1-\sqrt{3}\right)^4}}{3\sqrt{15}}=\frac{3\left(1-\sqrt{3}\right)^2}{3\sqrt{15}}=\frac{1-2\sqrt{3}+3}{\sqrt{15}}=\frac{4-2\sqrt{3}}{\sqrt{15}}\)
- \(=\frac{\sqrt{10}\left(12-8\sqrt{2}+7.15\sqrt{2}\right)}{\sqrt{10}}=12+97\sqrt{2}\)
- \(=\sqrt{\frac{x.x\sqrt{y}}{y}}=\sqrt{\frac{x^2}{\sqrt{y}}}=\frac{|x|}{\sqrt[4]{y}}\)
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a)\(\left(\sqrt{12}+\sqrt{75}+\sqrt{27}\right):\sqrt{15}\)
\(=\left(2\sqrt{3}+5\sqrt{3}+3\sqrt{3}\right):\sqrt{15}\)
\(=10\sqrt{3}:\sqrt{15}=\sqrt{300}:\sqrt{15}=\sqrt{20}=2\sqrt{5}\)
b) \(\frac{12\sqrt{50}-8\sqrt{200}+7\sqrt{450}}{\sqrt{10}}\)
\(=\frac{60\sqrt{2}-80\sqrt{2}+105\sqrt{2}}{\sqrt{10}}\)
\(=\frac{85\sqrt{2}}{10}=\frac{17\sqrt{2}}{2}\)
c)\(\frac{\sqrt{\frac{1}{7}}-\sqrt{\frac{16}{7}}+\sqrt{\frac{9}{7}}}{7}=\frac{\frac{1}{\sqrt{7}}-\frac{4}{\sqrt{7}}+\frac{3}{\sqrt{7}}}{7}=\frac{0}{7}=0\)
Bài 1:
a) Ta có: \(\frac{12\sqrt{50}-8\sqrt{200}+7\sqrt{450}}{\sqrt{10}}\)
\(=\frac{12\cdot\sqrt{5}\cdot\sqrt{10}-8\cdot\sqrt{20}\cdot\sqrt{10}+7\cdot\sqrt{45}\cdot\sqrt{10}}{\sqrt{10}}\)
\(=\frac{\sqrt{10}\left(12\sqrt{5}-8\sqrt{20}+7\sqrt{45}\right)}{\sqrt{10}}\)
\(=12\sqrt{5}-8\sqrt{20}+7\sqrt{45}\)
\(=\sqrt{5}\left(12-16+21\right)\)
\(=17\sqrt{5}\)
b) Ta có: \(\frac{\frac{\sqrt{1}}{7}-\sqrt{\frac{16}{7}}+\sqrt{\frac{9}{7}}}{\sqrt{7}}\)
\(=\left(\frac{1}{\sqrt{7}}-\frac{4}{\sqrt{7}}+\frac{3}{\sqrt{7}}\right)\cdot\frac{1}{\sqrt{7}}\)
\(=0\cdot\frac{1}{\sqrt{7}}=0\)
a) \(\sqrt{200}+2\sqrt{108}-\sqrt{98}+\frac{1}{3}\sqrt{\frac{81}{3}}-3\sqrt{75}\)
\(=10\sqrt{2}+12\sqrt{3}-7\sqrt{2}+\sqrt{3}-15\sqrt{3}\)
\(=3\sqrt{2}-2\sqrt{3}\)
b)\(\left(21\sqrt{\frac{1}{7}}+\frac{1}{2}\sqrt{112}-\frac{14}{3}\sqrt{\frac{9}{7}}+7\right):3\sqrt{7}\)
\(=\left(3\sqrt{7}+2\sqrt{7}-2\sqrt{7}+7\right):3\sqrt{7}\)
\(=\frac{\sqrt{7}\left(3+\sqrt{7}\right)}{3\sqrt{7}}=\frac{\sqrt{7}+3}{3}\)
c)\(\left(\sqrt{27}-\sqrt{125}+\sqrt{45}+\sqrt{12}\right)\left(\sqrt{75}+\sqrt{20}\right)\)
\(=\left(3\sqrt{3}-5\sqrt{5}+3\sqrt{5}+2\sqrt{3}\right)\left(5\sqrt{3}+2\sqrt{5}\right)\)
\(=\left(5\sqrt{3}-2\sqrt{5}\right)\left(5\sqrt{3}+2\sqrt{5}\right)\)
\(=75-20=55\)
d)\(\left(\frac{3}{\sqrt{6}-3}-\frac{3}{\sqrt{6}+3}\right).\frac{3-\sqrt{3}}{2-2\sqrt{3}}-\frac{\sqrt{28-6\sqrt{3}}}{1}\)
\(=\frac{3\left(\sqrt{6}+3\right)-3\left(\sqrt{6}-3\right)}{-3}.\frac{3-\sqrt{3}}{2-2\sqrt{3}}-\sqrt{\left(3\sqrt{3}-1\right)^2}\)
\(=\frac{-6\left(3-\sqrt{3}\right)}{2-2\sqrt{3}}-\left(3\sqrt{3}-1\right)\left(do3\sqrt{3}>1\right)\)
\(=\frac{6\sqrt{3}-18}{2-2\sqrt{3}}-\frac{8\sqrt{3}-20}{2-2\sqrt{3}}\)
\(=\frac{6\sqrt{3}-18-8\sqrt{3}+20}{2-2\sqrt{3}}=\frac{2-2\sqrt{3}}{2-2\sqrt{3}}=1\)