Đáp án:
\[2 + \sqrt 3 \]
Giải thích các bước giải:
Ta có:
\(\begin{array}{l}
\dfrac{{13}}{{5 - 2\sqrt 3 }} + \dfrac{{3 - \sqrt 3 }}{{\sqrt 3 + 1}} - \dfrac{9}{{\sqrt 3 }}\\
= \dfrac{{13.\left( {5 + 2\sqrt 3 } \right)}}{{\left( {5 - 2\sqrt 3 } \right)\left( {5 + 2\sqrt 3 } \right)}} + \dfrac{{\sqrt 3 \left( {\sqrt 3 - 1} \right)}}{{\sqrt 3 + 1}} - \dfrac{{{3^2}}}{{\sqrt 3 }}\\
= \dfrac{{13.\left( {5 + 2\sqrt 3 } \right)}}{{{5^2} - {{\left( {2\sqrt 3 } \right)}^2}}} + \dfrac{{\sqrt 3 .{{\left( {\sqrt 3 - 1} \right)}^2}}}{{\left( {\sqrt 3 + 1} \right)\left( {\sqrt 3 - 1} \right)}} - \dfrac{{3.{{\sqrt 3 }^2}}}{{\sqrt 3 }}\\
= \dfrac{{13.\left( {5 + 2\sqrt 3 } \right)}}{{25 - 12}} + \dfrac{{\sqrt 3 .\left( {3 - 2\sqrt 3 + 1} \right)}}{{3 - 1}} - 3.\sqrt 3 \\
= \dfrac{{13.\left( {5 + 2\sqrt 3 } \right)}}{{13}} + \dfrac{{\sqrt 3 .\left( {4 - 2\sqrt 3 } \right)}}{2} - 3\sqrt 3 \\
= \left( {5 + 2\sqrt 3 } \right) + \sqrt 3 .\left( {2 - \sqrt 3 } \right) - 3\sqrt 3 \\
= 5 + 2\sqrt 3 + 2\sqrt 3 - 3 - 3\sqrt 3 \\
= 2 + \sqrt 3
\end{array}\)
