Đáp án:
$\begin{array}{l}
a)\dfrac{{\sqrt {15} - \sqrt {12} }}{{\sqrt 5 - 2}} - \dfrac{{6 + 2\sqrt 6 }}{{\sqrt 3 + \sqrt 2 }}\\
= \dfrac{{\sqrt 3 \left( {\sqrt 5 - 2} \right)}}{{\sqrt 5 - 2}} - \dfrac{{\sqrt {12} .\left( {\sqrt 3 + \sqrt 2 } \right)}}{{\sqrt 3 + \sqrt 2 }}\\
= \sqrt 3 - \sqrt {12} \\
= \sqrt 3 - 2\sqrt 3 \\
= - \sqrt 3 \\
b)\left( {\sqrt 3 - \sqrt 2 } \right).\sqrt {5 + 2\sqrt 6 } \\
= \left( {\sqrt 3 - \sqrt 2 } \right).\sqrt {{{\left( {\sqrt 3 + \sqrt 2 } \right)}^2}} \\
= \left( {\sqrt 3 - \sqrt 2 } \right)\left( {\sqrt 3 + \sqrt 2 } \right)\\
= 3 - 2\\
= 1\\
c)\dfrac{1}{{1 + \sqrt 2 }} + \dfrac{1}{{\sqrt 2 + \sqrt 3 }} + ... + \dfrac{1}{{\sqrt {99} + \sqrt {100} }}\\
= \dfrac{{\sqrt 2 - 1}}{{2 - 1}} + \dfrac{{\sqrt 3 - \sqrt 2 }}{{3 - 2}} + ... + \dfrac{{\sqrt {100} - \sqrt {99} }}{{100 - 99}}\\
= \sqrt 2 - 1 + \sqrt 3 - \sqrt 2 + ... + \sqrt {100} - \sqrt {99} \\
= \sqrt {100} - 1\\
= 10 - 1\\
= 9\\
d)\left( {\dfrac{{\sqrt x - \sqrt y }}{{x - y}} + \dfrac{{\sqrt {xy} }}{{\sqrt x + \sqrt y }}} \right):\dfrac{{\sqrt {xy} + 1}}{{\sqrt x + \sqrt y }}\\
= \left( {\dfrac{1}{{\sqrt x + \sqrt y }} + \dfrac{{\sqrt {xy} }}{{\sqrt x + \sqrt y }}} \right).\dfrac{{\sqrt x + \sqrt y }}{{\sqrt {xy} + 1}}\\
= \dfrac{{\sqrt {xy} + 1}}{{\sqrt x + \sqrt y }}.\dfrac{{\sqrt x + \sqrt y }}{{\sqrt {xy} + 1}}\\
= 1\\
e)\left( {\sqrt 6 + \sqrt 2 } \right)\left( {\sqrt 3 - 2} \right)\sqrt {\sqrt 3 + 2} \\
= \left( {\sqrt 3 + 1} \right).\left( {\sqrt 3 - 2} \right).\sqrt 2 .\sqrt {\sqrt 3 + 2} \\
= \left( {\sqrt 3 + 1} \right).\dfrac{1}{2}.\left( {2\sqrt 3 - 4} \right).\sqrt {2\sqrt 3 + 4} \\
= - \dfrac{1}{2}\left( {\sqrt 3 + 1} \right).{\left( {\sqrt 3 - 1} \right)^2}.\sqrt {{{\left( {\sqrt 3 + 1} \right)}^2}} \\
= - \dfrac{1}{2}{\left( {\sqrt 3 - 1} \right)^2}.\left( {\sqrt 3 + 1} \right).\left( {\sqrt 3 + 1} \right)\\
= - \dfrac{1}{2}.{\left[ {\left( {\sqrt 3 - 1} \right).\left( {\sqrt 3 + 1} \right)} \right]^2}\\
= - \dfrac{1}{2}.{\left( {3 - 1} \right)^2}\\
= - 2\\
g)\left( {4 + \sqrt {15} } \right).\left( {\sqrt {10} - \sqrt 6 } \right).\sqrt {4 - \sqrt {15} } \\
= \dfrac{1}{2}.\left( {8 + 2\sqrt {15} } \right).\left( {\sqrt 5 - \sqrt 3 } \right).\sqrt 2 .\sqrt {4 - \sqrt {15} } \\
= \dfrac{1}{2}.{\left( {\sqrt 5 + \sqrt 3 } \right)^2}.\left( {\sqrt 5 - \sqrt 3 } \right).\sqrt {{{\left( {\sqrt 5 - \sqrt 3 } \right)}^2}} \\
= \dfrac{1}{2}.{\left( {\sqrt 5 + \sqrt 3 } \right)^2}.{\left( {\sqrt 5 - \sqrt 3 } \right)^2}\\
= \dfrac{1}{2}.{\left( {5 - 3} \right)^2}\\
= 2
\end{array}$
