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) \(2\sqrt {\frac{2}{3}}  - 4\sqrt {\frac{3}{2}} \)\( = 2\frac{{\sqrt 2 }}{{\sqrt 3 }} - 4\frac{{\sqrt 3 }}{{\sqrt 2 }}\)\( = 2.\frac{{\sqrt 6 }}{3} - 4.\frac{{\sqrt 6 }}{2}\)\( = \sqrt 6 \left( {\frac{2}{3} - 2} \right)\)\( = \frac{{ - 4\sqrt 6 }}{3}.\) b) \(\frac{{5\sqrt {48}  - 3\sqrt {27}  + 2\sqrt {12} }}{{\sqrt 3 }}\)\( = \frac{{5\sqrt {16.3}  - 3\sqrt {9.3}  + 2\sqrt {4.3} }}{{\sqrt 3 }}\)\( = \frac{{\sqrt 3 .\left( {5\sqrt {16}  - 3\sqrt 9  + 2\sqrt 4 } \right)}}{{\sqrt 3 }}\)\( = 5.4 - 3.3 + 2.2\)\( = 20 - 9 + 4\)\( = 15\) c) \(\frac{1}{{3 + 2\sqrt 2 }} + \frac{{4\sqrt 2  - 4}}{{2 - \sqrt 2 }}\)\( = \frac{{3 - 2\sqrt 2 }}{{\left( {3 + 2\sqrt 2 } \right)\left( {3 - 2\sqrt 2 } \right)}} + \frac{{4\left( {\sqrt 2  - 1} \right)}}{{\sqrt 2 \left( {\sqrt 2  - 1} \right)}}\)\( = \frac{{3 - 2\sqrt 2 }}{{9 - 8}} + \frac{4}{{\sqrt 2 }}\)\( = 3 - 2\sqrt 2  + \frac{{4\sqrt 2 }}{2}\) \( = 3 - 2\sqrt 2  + 2\sqrt 2 \)\( = 3\)