Đáp án:
c. \(\sqrt 3 - 1\)
Giải thích các bước giải:
\(\begin{array}{l}
a.\left( {4 + \sqrt 5 } \right).\sqrt 2 \left( {\sqrt 5 - \sqrt 3 } \right)\sqrt {4 - \sqrt {15} } \\
= \left( {4 + \sqrt 5 } \right)\left( {\sqrt 5 - \sqrt 3 } \right)\sqrt {8 - 2\sqrt {15} } \\
= \left( {4 + \sqrt 5 } \right)\left( {\sqrt 5 - \sqrt 3 } \right)\sqrt {5 - 2\sqrt 5 .\sqrt 3 + 3} \\
= \left( {4 + \sqrt 5 } \right)\left( {\sqrt 5 - \sqrt 3 } \right)\sqrt {{{\left( {\sqrt 5 - \sqrt 3 } \right)}^2}} \\
= \left( {4 + \sqrt 5 } \right){\left( {\sqrt 5 - \sqrt 3 } \right)^2}\\
= \left( {4 + \sqrt 5 } \right)\left( {8 - 2\sqrt {15} } \right)\\
= 16 - 8\sqrt {15} + 8\sqrt 5 - 10\sqrt 3 \\
b.\sqrt {2 - \sqrt 3 } .\sqrt 2 \left( {\sqrt 3 + 1} \right)\\
= \sqrt {4 - 2\sqrt 3 } .\left( {\sqrt 3 + 1} \right)\\
= \left( {\sqrt 3 + 1} \right).\sqrt {3 - 2\sqrt 3 .1 + 1} \\
= \left( {\sqrt 3 + 1} \right)\sqrt {{{\left( {\sqrt 3 - 1} \right)}^2}} \\
= \left( {\sqrt 3 + 1} \right)\left( {\sqrt 3 - 1} \right) = 3 - 1 = 2\\
c.\sqrt {6 - 2\sqrt {\sqrt 2 + 2\sqrt 3 + \sqrt {18 - 8\sqrt 2 } } } \\
= \sqrt {6 - 2\sqrt {\sqrt 2 + 2\sqrt 3 + \sqrt {16 - 2.4.\sqrt 2 + 2} } } \\
= \sqrt {6 - 2\sqrt {\sqrt 2 + 2\sqrt 3 + \sqrt {{{\left( {4 - \sqrt 2 } \right)}^2}} } } \\
= \sqrt {6 - 2\sqrt {\sqrt 2 + 2\sqrt 3 + 4 - \sqrt 2 } } \\
= \sqrt {6 - 2\sqrt {4 + 2\sqrt 3 } } \\
= \sqrt {6 - 2\sqrt {{{\left( {\sqrt 3 + 1} \right)}^2}} } \\
= \sqrt {6 - 2\left( {\sqrt 3 + 1} \right)} \\
= \sqrt {6 - 2\sqrt 3 - 2} \\
= \sqrt {4 - 2\sqrt 3 } = \sqrt {{{\left( {\sqrt 3 - 1} \right)}^2}} \\
= \sqrt 3 - 1
\end{array}\)
