Thực hiện phép tính (a) (căn 45−căn 20+căn 5):căn 6

a) \[\left( {\sqrt {45} - \sqrt {20} + \sqrt 5 } \right):\sqrt 6 \]

\[ = \left( {\sqrt 9 \sqrt 5 - \sqrt 4 \sqrt 5 + \sqrt 1 \sqrt 5 } \right):\sqrt 6 \]

\[ = \left( {3\sqrt 5 - 2\sqrt 5 + 1\sqrt 5 } \right):\sqrt 6 \]

\[ = \left[ {\left( {3 - 2 + 1} \right)\sqrt 5 } \right]:\sqrt 6 \]

\[ = 2\sqrt 5 :\sqrt 6 \]

\[ = \frac{{2\sqrt 5 }}{{\sqrt 6 }} = \frac{{\sqrt 2 .\sqrt 2 .\sqrt 5 }}{{\sqrt 2 .\sqrt 3 }} = \frac{{\sqrt {10} }}{{\sqrt 3 }}\]

b) \[\left( {2\sqrt 6 - 4\sqrt 3 + 5\sqrt 2 - \frac{1}{4}\sqrt 8 } \right).3\sqrt 6 \]

\[ = \left[ {\left( {2\sqrt 2 \sqrt 3 - 4\sqrt 3 } \right) + \left( {5\sqrt 2 - \frac{1}{4}2\sqrt 2 } \right)} \right].3\sqrt 6 \]

\[ = \left[ {\left( { - 4 + 2\sqrt 2 } \right)\sqrt 3 + \frac{9}{2}\sqrt 2 } \right].3\sqrt 6 \]

\[ = \left( { - 4 + 2\sqrt 2 } \right)\sqrt 3 .3\sqrt 6 + \frac{9}{2}\sqrt 2 .3\sqrt 6 \]

\[ = - 36\sqrt 2 + 36 + 27\sqrt 3 \]

c) \[\left( {\sqrt 6 + 2} \right)\left( {\sqrt 3 - \sqrt 2 } \right)\]

\[ = \sqrt 6 .\sqrt 3 - \sqrt 6 .\sqrt 2 + 2\sqrt 3 - 2\sqrt 2 \]

\[ = 3\sqrt 2 - 2\sqrt 3 + 2\sqrt 3 - 2\sqrt 2 \]

\[ = \left( {3\sqrt 2 - 2\sqrt 2 } \right)\left( { - 2\sqrt 3 + 2\sqrt 3 } \right)\]

\[ = \sqrt 2 \]

d) \[\left( {\frac{1}{3}\sqrt {\frac{1}{2}} - \frac{2}{3}\sqrt {\frac{3}{2}} + \frac{2}{7}\sqrt {\frac{1}{6}} } \right):\left( {\frac{2}{7}\sqrt {\frac{1}{8}} } \right)\]

\[ = \left( {\frac{1}{3} \cdot \frac{{\sqrt 1 }}{{\sqrt 2 }} - \frac{2}{3} \cdot \frac{{\sqrt 3 }}{{\sqrt 2 }} + \frac{2}{7} \cdot \frac{{\sqrt 1 }}{{\sqrt 6 }}} \right):\left( {\frac{2}{7} \cdot \frac{{\sqrt 1 }}{{\sqrt 8 }}} \right)\]

\[ = \left( {\frac{1}{{3\sqrt 2 }} - \frac{{2\sqrt 3 }}{{3\sqrt 2 }} + \frac{{\sqrt 2 }}{{7\sqrt 3 }}} \right):\frac{1}{{7\sqrt 2 }}\]

\[ = \left( {\frac{1}{{3\sqrt 2 }} \cdot \frac{{7\sqrt 2 }}{1}} \right) - \left( {\frac{{2\sqrt 3 }}{{3\sqrt 2 }} \cdot \frac{{7\sqrt 2 }}{1}} \right) + \left( {\frac{{\sqrt 2 }}{{7\sqrt 3 }} \cdot \frac{{7\sqrt 2 }}{1}} \right)\]

\[ = \frac{7}{3} - \frac{{14\sqrt 3 }}{3} + \frac{{2\sqrt 3 }}{3} = \frac{{7 - 12\sqrt 3 }}{3}\]