Rút gọn các biểu thức:
a) \(4\sqrt {24} + \sqrt 6 - 2\sqrt {54} \)
b) \(2\sqrt {45} - \sqrt {125} - \frac{{15}}{{\sqrt 5 }}\)
c) \(\sqrt 8 - \sqrt {27} - \sqrt {32} + \sqrt {75} \)
d) \(\left( {2 - \sqrt {10} } \right)\left( {\sqrt 2 - \sqrt 5 } \right)\)
Dựa vào: \(\frac{{\sqrt a }}{{\sqrt b }} = \frac{{\sqrt a .\sqrt b }}{{{{\left( {\sqrt b } \right)}^2}}} = \frac{{\sqrt {ab} }}{b}(a \ge 0,b > 0)\)
\(\sqrt {\frac{a}{b}} = \sqrt {\frac{{ab}}{{{b^2}}}} = \frac{{\sqrt {ab} }}{b}(a \ge 0,b > 0)\)
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a) \(4\sqrt {24} + \sqrt 6 - 2\sqrt {54} \)
\(= 8\sqrt 6 + \sqrt 6 - 6\sqrt 6 = 3\sqrt 6 .\)
b) \(2\sqrt {45} - \sqrt {125} - \frac{{15}}{{\sqrt 5 }}\)
\(= 5\sqrt 5 - 5\sqrt 5 - 3\sqrt 5 = - 2\sqrt 5 .\)
c) \(\sqrt 8 - \sqrt {27} - \sqrt {32} + \sqrt {75} \)
\(= 2\sqrt 2 - 3\sqrt 3 - 4\sqrt 2 + 5\sqrt 3 \\= (2 - 4)\sqrt 2 + ( - 3 + 5)\sqrt 3 \\ = - 2\sqrt 2 + 2\sqrt 3 .\)
d) \(\left( {2 - \sqrt {10} } \right)\left( {\sqrt 2 - \sqrt 5 } \right) \)
\(= 2\sqrt 2 - 2\sqrt 5 - \sqrt {10} .\sqrt 2 + \sqrt {10} .\sqrt 5 \\ = 2\sqrt 2 - 2\sqrt 5 - 2\sqrt 5 + 5\sqrt 2 \\ = 7\sqrt 2 - 4\sqrt 5 .\)