Đáp án:
\(\begin{array}{l}
1)1\\
2)7\\
3)\dfrac{{2\sqrt 3 - 1}}{{\sqrt {3 - \sqrt 3 } }}\\
4)\dfrac{{\sqrt {21} }}{7}\\
5)2 + \sqrt 2 - \sqrt 3 - \sqrt 6
\end{array}\)
Giải thích các bước giải:
\(\begin{array}{l}
1)\sqrt {\left( {\sqrt 2 - 1} \right)\left( {\sqrt 2 + 1} \right)} = \sqrt {2 - 1} = 1\\
2)\sqrt {9 - 2.3.\sqrt 2 + 2} .\sqrt {9 + 2.3.\sqrt 2 + 2} \\
= \sqrt {{{\left( {3 - \sqrt 2 } \right)}^2}} .\sqrt {{{\left( {3 + \sqrt 2 } \right)}^2}} \\
= \left( {3 - \sqrt 2 } \right).\left( {3 + \sqrt 2 } \right)\\
= 9 - 2 = 7\\
3)\sqrt {12 - 2.2\sqrt 3 .1 + 1} .\dfrac{1}{{\sqrt {3 - \sqrt 3 } }}\\
= \dfrac{{\sqrt {{{\left( {2\sqrt 3 - 1} \right)}^2}} }}{{\sqrt {3 - \sqrt 3 } }}\\
= \dfrac{{2\sqrt 3 - 1}}{{\sqrt {3 - \sqrt 3 } }}\\
4)\dfrac{{\sqrt 3 \left( {\sqrt 5 - \sqrt 2 } \right)}}{{\sqrt 7 \left( {\sqrt 5 - \sqrt 2 } \right)}}\\
= \dfrac{{\sqrt {21} }}{7}\\
5)\dfrac{{2\sqrt 5 \left( {\sqrt 3 - \sqrt 2 } \right) + \sqrt 3 \left( {\sqrt 2 - \sqrt 3 } \right)}}{{2\sqrt 5 \left( {1 - \sqrt 2 } \right) - \sqrt 3 \left( {1 - \sqrt 2 } \right)}}\\
= \dfrac{{2\sqrt 5 \left( {\sqrt 3 - \sqrt 2 } \right) - \sqrt 3 \left( {\sqrt 3 - \sqrt 2 } \right)}}{{2\sqrt 5 \left( {1 - \sqrt 2 } \right) - \sqrt 3 \left( {1 - \sqrt 2 } \right)}}\\
= \dfrac{{\left( {\sqrt 3 - \sqrt 2 } \right)\left( {2\sqrt 5 - \sqrt 3 } \right)}}{{\left( {1 - \sqrt 2 } \right)\left( {2\sqrt 5 - \sqrt 3 } \right)}}\\
= \dfrac{{\sqrt 3 - \sqrt 2 }}{{1 - \sqrt 2 }}\\
= \dfrac{{\left( {\sqrt 3 - \sqrt 2 } \right)\left( {1 + \sqrt 2 } \right)}}{{1 - 2}}\\
= \dfrac{{\sqrt 6 + \sqrt 3 - \sqrt 2 - 2}}{{ - 1}}\\
= 2 + \sqrt 2 - \sqrt 3 - \sqrt 6
\end{array}\)
