9 bài tập Rút gọn biểu thức (có lời giải)

a) Ta có \(2\sqrt {3x}  - 4\sqrt {3x}  + 27 - 3\sqrt {3x}  = (2 - 4 - 3)\sqrt {3x}  + 27 =  - 5\sqrt 3 x + 27\).

b) Ta có \(3\sqrt {2x}  - 5\sqrt {8x}  + 7\sqrt {18x}  + 28 = 3\sqrt {2x}  - 10\sqrt {2x}  + 21\sqrt {2x}  + 28\)

                                              \( = (3 - 10 + 21)\sqrt {2x}  + 28 = 14\sqrt {2x}  + 28\).

a) Ta có \(\frac{2}{{{x^2} - {y^2}}}\sqrt {\frac{{3{{(x + y)}^2}}}{2}}  = \frac{{|x + y|}}{{{x^2} - {y^2}}}\sqrt {\frac{{4.3}}{2}}  = \frac{{x + y}}{{(x - y)(x + y)}}.\sqrt 6  = \frac{{\sqrt 6 }}{{x - y}}\) (vì \(x + y > 0\)) .

b) \(\frac{2}{{2a - 1}}\sqrt {5{a^2}\left( {1 - 4a + 4{a^2}} \right)}  = \frac{2}{{2a - 1}}\sqrt {5{a^2}{{(1 - 2a)}^2}} \)

         \( = \frac{2}{{2a - 1}}|a|.|1 - 2a|\sqrt 5  = \frac{{2\sqrt 5 }}{{2a - 1}}.a.(2a - 1)\quad \left( {{\rm{ do }}a > \frac{1}{2}} \right) = 2\sqrt 5 {\rm{a}}.\)

a). \[\sqrt {18{{\left( {\sqrt 2  - \sqrt 3 } \right)}^2}}  = \sqrt {2.9{{\left( {\sqrt 2  - \sqrt 3 } \right)}^2}}  = 3\,\left| {\sqrt 2  - \sqrt 3 } \right|\sqrt 2  = 3\left( {\sqrt 3  - \sqrt 2 } \right)\sqrt 2  = 3\sqrt 6  - 6.\]

b). \(ab\sqrt {1 + \frac{1}{{{a^2}{b^2}}}}  = ab\sqrt {\frac{{{a^2}{b^2} + 1}}{{{a^2}{b^2}}}}  = \frac{{ab}}{{\,\left| {ab} \right|}}\sqrt {{a^2}{b^2} + 1} \)\(\, = \left\{ \begin{array}{l}\sqrt {{a^2}{b^2} + 1} \,\,\,neu\,ab > 0\,\,\,\\ - \sqrt {{a^2}{b^2} + 1} \,\,\,neu\,\,ab < 0.\end{array} \right.\)

c). \(\sqrt {\frac{a}{{{b^3}}} + \frac{a}{{{b^4}}}}  = \sqrt {\frac{{ab + a}}{{{b^4}}}}  = \frac{{\sqrt {ab + a} }}{{{b^2}}}.\)

d). \(\frac{{a + \sqrt {ab} }}{{\sqrt a  + \sqrt b }} = \frac{{\sqrt a \left( {\sqrt a  + \sqrt b } \right)}}{{\sqrt a  + \sqrt b }} = \sqrt a .\)

Cách khác: \(\frac{{a + \sqrt {ab} }}{{\sqrt a  + \sqrt b }} = \frac{{\left( {a + \sqrt {ab} } \right)\left( {\sqrt a  - \sqrt b } \right)}}{{a - b}} = \frac{{\sqrt a \left( {\sqrt a  + \sqrt b } \right)\left( {\sqrt a  - \sqrt b } \right)}}{{a - b}} = \frac{{\sqrt a \left( {a - b} \right)}}{{a - b}} = \sqrt a .\)

Ta có: \[\frac{{2 + \sqrt 2 }}{{1 + \sqrt 2 }} = \frac{{\sqrt 2 \left( {1 + \sqrt 2 } \right)}}{{1 + \sqrt 2 }} = \sqrt 2 ;\]           \[\frac{{\sqrt {15}  - \sqrt 5 }}{{1 - \sqrt 3 }} = \frac{{\sqrt 5 \left( {\sqrt 3  - 1} \right)}}{{1 - \sqrt 3 }} =  - \sqrt 5 ;\]

                   \[\frac{{2\sqrt 3  - \sqrt 6 }}{{\sqrt 8  - 2}} = \frac{{\sqrt 6 \left( {\sqrt 2  - 1} \right)}}{{2\left( {\sqrt 2  - 1} \right)}} = \frac{{\sqrt 6 }}{2};\]              \[\frac{{a - \sqrt a }}{{1 - \sqrt a }} = \frac{{\sqrt a \left( {\sqrt a  - 1} \right)}}{{1 - ra}} =  - \sqrt a ;\]

\[\frac{{p - 2\sqrt p }}{{\sqrt p  - 2}} = \frac{{\sqrt p \left( {\sqrt p  - 2} \right)}}{{\sqrt p  - 2}} = \sqrt p .\]

a) \[5\sqrt a  - 4b\sqrt {25{a^3}}  + 5\sqrt {16a{b^2}}  - 2\sqrt {9a}  = 5\sqrt a  - 20ab\sqrt a  + 20ab\sqrt a  - 6\sqrt a  =  - \sqrt a .\]

a) \[B = 4\sqrt {x + 1}  - 3\sqrt {x + 1}  + 2\sqrt {x + 1}  + \sqrt {x + 1}  = 4\sqrt {x + 1} .\]

b) \[B = 16 \Leftrightarrow 4\sqrt {x + 1}  = 16 \Leftrightarrow \sqrt {x + 1}  = 4 \Leftrightarrow x + 1 = 16 \Leftrightarrow x = 15.\]