Đề bài

Rút gọn các biểu thức sau: a) \(2\sqrt {\frac{2}{3}}  - 4\sqrt {\frac{3}{2}} \); b) \(\frac{{5\sqrt {48}  - 3\sqrt {27}  + 2\sqrt {12} }}{{\sqrt 3 }}\); c) \(\frac{1}{{3 + 2\sqrt 2 }} + \frac{{4\sqrt 2  - 4}}{{2 - \sqrt 2 }}\).

Lời giải chi tiết

a) \(2\sqrt {\frac{2}{3}}  - 4\sqrt {\frac{3}{2}} \) \(= 2\sqrt {\frac{{2.3}}{{{3^2}}}}  - 4.\sqrt {\frac{{3.2}}{{{2^2}}}} \\= \frac{2}{3}\sqrt 6  - \frac{4}{2}\sqrt 6  =  - \frac{4}{3}\sqrt 6;\) b) \(\frac{{5\sqrt {48}  - 3\sqrt {27}  + 2\sqrt {12} }}{{\sqrt 3 }} \) \(= \frac{{5\sqrt {48} }}{{\sqrt 3 }} - \frac{{3\sqrt {27} }}{{\sqrt 3 }} + \frac{{2\sqrt {12} }}{{\sqrt 3 }}\\ = 5\sqrt {\frac{{48}}{3}}  - 3\sqrt {\frac{{27}}{3}}  + 2\sqrt {\frac{{12}}{3}}  \\= 5.4 - 3.3 + 2.2 = 15\) c) \(\frac{1}{{3 + 2\sqrt 2 }} + \frac{{4\sqrt 2  - 4}}{{2 - \sqrt 2 }} \) \(= \frac{{3 - 2\sqrt 2 }}{{\left( {3 + 2\sqrt 2 } \right)\left( {3 - 2\sqrt 2 } \right)}} + \frac{{4\left( {\sqrt 2  - 1} \right)}}{{\sqrt 2 \left( {\sqrt 2  - 1} \right)}}\\ = \frac{{3 - 2\sqrt 2 }}{{{3^2} - {{\left( {2\sqrt 2 } \right)}^2}}} + \frac{4}{{\sqrt 2 }} \\= 3 - 2\sqrt 2  + 2\sqrt 2  = 3\)