Đáp án:
$\begin{array}{l}
a)\left( {2\sqrt 6 - 4\sqrt 3 + 5\sqrt 2 - \frac{1}{4}\sqrt 8 } \right).\sqrt 6 \\
= \left( {2\sqrt 6 - 4\sqrt 3 + 5\sqrt 2 - \frac{1}{2}\sqrt 2 } \right).\sqrt 6 \\
= 2.6 - 4\sqrt 3 .\sqrt 6 + \frac{9}{2}\sqrt 2 .\sqrt 6 \\
= 12 - 12\sqrt 2 + 9\sqrt 3 \\
b)\left( {20\sqrt {300} - 15\sqrt {675} + 5\sqrt {75} } \right):\sqrt {15} \\
= \left( {20.10\sqrt 3 - 15.15\sqrt 3 + 5.5\sqrt 3 } \right):\sqrt {15} \\
= 0:\sqrt {15} \\
= 0\\
c){\left( {\sqrt 2 + 1} \right)^2} - {\left( {1 - \sqrt 2 } \right)^2}\\
= \left( {\sqrt 2 + 1 - 1 + \sqrt 2 } \right)\left( {\sqrt 2 + 1 + 1 - \sqrt 2 } \right)\\
= 2\sqrt 2 .2\\
= 4\sqrt 2 \\
d){\left( {\sqrt 2 - \sqrt 6 } \right)^2} + {\left( {\sqrt 3 - 2} \right)^2}\\
= 2 - 2.\sqrt 2 .\sqrt 6 + 6 + 3 - 4\sqrt 3 + 4\\
= 8 - 4\sqrt 3 + 7 - 4\sqrt 3 \\
= 15 - 8\sqrt 3 \\
e){\left( {\sqrt 7 + 1 - \sqrt {14} } \right)^2} - {\left( {\sqrt 2 + 1} \right)^2} + 2\sqrt {14} \\
= {\left( {\sqrt 7 + 1} \right)^2} - 2.\left( {\sqrt 7 + 1} \right).\sqrt {14} + 14\\
- 2 - 2\sqrt 2 - 1 + 2\sqrt {14} \\
= 8 + 2\sqrt 7 - 2.\sqrt 7 .\sqrt {14} - 2\sqrt {14} + 14\\
- 3 - 2\sqrt 2 + 2\sqrt {14} \\
= 8 - 2\sqrt 7 - 14\sqrt 2 + 14 - 3 - 2\sqrt 2 \\
= 19 - 2\sqrt 7 - 16\sqrt 2 \\
f)\left( {3 - \sqrt 5 } \right)\left( {3\sqrt {10} - \sqrt 2 } \right)\left( {3 + \sqrt 5 } \right)\\
= \left( {3 - \sqrt 5 } \right)\left( {3 + \sqrt 5 } \right).\left( {3\sqrt {10} - \sqrt 2 } \right)\\
= \left( {9 - 5} \right).\left( {3\sqrt {10} - \sqrt 2 } \right)\\
= 12\sqrt {10} - 4\sqrt 2
\end{array}$
