Đáp án:
\[\sqrt 5 \]
Giải thích các bước giải:
Ta có:
\(\begin{array}{l}
\dfrac{{5 + \sqrt 5 }}{{\sqrt 5 + 2}} + \dfrac{{\sqrt 5 }}{{\sqrt 5 - 1}} - \dfrac{{3\sqrt 5 }}{{3 + \sqrt 5 }}\\
= \dfrac{{\left( {5 + \sqrt 5 } \right)\left( {\sqrt 5 - 2} \right)}}{{\left( {\sqrt 5 + 2} \right)\left( {\sqrt 5 - 2} \right)}} + \dfrac{{\sqrt 5 \left( {\sqrt 5 + 1} \right)}}{{\left( {\sqrt 5 - 1} \right)\left( {\sqrt 5 + 1} \right)}} - \dfrac{{3\sqrt 5 \left( {3 - \sqrt 5 } \right)}}{{\left( {3 + \sqrt 5 } \right)\left( {3 - \sqrt 5 } \right)}}\\
= \dfrac{{5\sqrt 5 - 10 + {{\sqrt 5 }^2} - 2\sqrt 5 }}{{{{\sqrt 5 }^2} - {2^2}}} + \dfrac{{{{\sqrt 5 }^2} + \sqrt 5 }}{{{{\sqrt 5 }^2} - {1^2}}} - \dfrac{{9\sqrt 5 - 3.{{\sqrt 5 }^2}}}{{{3^2} - {{\sqrt 5 }^2}}}\\
= \dfrac{{5\sqrt 5 - 10 + 5 - 2\sqrt 5 }}{{5 - 4}} + \dfrac{{5 + \sqrt 5 }}{{5 - 1}} - \dfrac{{9\sqrt 5 - 15}}{{9 - 5}}\\
= \dfrac{{3\sqrt 5 - 5}}{1} + \dfrac{{5 + \sqrt 5 }}{4} - \dfrac{{9\sqrt 5 - 15}}{4}\\
= 3\sqrt 5 - 5 + \dfrac{{5 + \sqrt 5 - 9\sqrt 5 + 15}}{4}\\
= 3\sqrt 5 - 5 + \dfrac{{20 - 8\sqrt 5 }}{4}\\
= 3\sqrt 5 - 5 + 5 - 2\sqrt 5 \\
= \sqrt 5
\end{array}\)
