Đề bài

Rút gọn các biểu thức sau: a) \(\frac{{5 + 3\sqrt 5 }}{{\sqrt 5 }} - \frac{1}{{\sqrt 5  - 2}};\) b) \(\sqrt {{{\left( {\sqrt 7  - 2} \right)}^2}}  - \sqrt {63}  + \frac{{\sqrt {56} }}{{\sqrt 2 }};\) c) \(\frac{{\sqrt {{{\left( {\sqrt 3  + \sqrt 2 } \right)}^2}}  + \sqrt {{{\left( {\sqrt 3  - \sqrt 2 } \right)}^2}} }}{{2\sqrt {12} }};\) d) \(\frac{{\sqrt[3]{{{{\left( {\sqrt 2  + 1} \right)}^3}}} - 1}}{{\sqrt {50} }}.\)

Lời giải chi tiết

a) \(\frac{{5 + 3\sqrt 5 }}{{\sqrt 5 }} - \frac{1}{{\sqrt 5  - 2}}\) \(\begin{array}{l} = \frac{{\sqrt 5\left( {\sqrt 5 + 3 } \right) }}{{\sqrt 5 }} - \frac{{\sqrt 5  + 2}}{{\left( {\sqrt 5  - 2} \right)\left( {\sqrt 5  + 2} \right)}}\\ =\sqrt 5  + 3  - \frac{{\sqrt 5  + 2}}{{5 - 4}}\end{array}\) \(\begin{array}{l} = \sqrt 5  + 3 - \left( {\sqrt 5  + 2} \right)\\ = 1\end{array}\) b) \(\sqrt {{{\left( {\sqrt 7  - 2} \right)}^2}}  - \sqrt {63}  + \frac{{\sqrt {56} }}{{\sqrt 2 }}\) \(\begin{array}{l} = \left| {\sqrt 7  - 2} \right| - \sqrt {9.7}  + \frac{{\sqrt {2.28} }}{{\sqrt 2 }}\\ = \sqrt 7  - 2 - 3\sqrt 7  + \sqrt {28} \\ =  - 2 - 2\sqrt 7  + \sqrt {4.7} \end{array}\) \(\begin{array}{l} =  - 2 - 2\sqrt 7  + 2\sqrt 7 \\ =  - 2\end{array}\) c) \(\frac{{\sqrt {{{\left( {\sqrt 3  + \sqrt 2 } \right)}^2}}  + \sqrt {{{\left( {\sqrt 3  - \sqrt 2 } \right)}^2}} }}{{2\sqrt {12} }}\) \(\begin{array}{l} = \frac{{\left| {\sqrt 3  + \sqrt 2 } \right| + \left| {\sqrt 3  - \sqrt 2 } \right|}}{{2\sqrt {4.3} }}\\ = \frac{{\sqrt 3  + \sqrt 2  + \sqrt 3  - \sqrt 2 }}{{4\sqrt 3 }}\\ = \frac{{2\sqrt 3 }}{{4\sqrt 3 }}\\ = \frac{1}{2}\end{array}\) d) \(\frac{{\sqrt[3]{{{{\left( {\sqrt 2  + 1} \right)}^3}}} - 1}}{{\sqrt {50} }}\) \(\begin{array}{l} = \frac{{\sqrt 2  + 1 - 1}}{{\sqrt {25.2} }}\\ = \frac{{\sqrt 2 }}{{5\sqrt 2 }}\\ = \frac{1}{5}\end{array}\)