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

b) $5a\sqrt {64a{b^3}} - \sqrt 3 \sqrt {12{a^3}{b^3}} + 2ab\sqrt {9ab} - 5b\sqrt {81{a^3}b}$

$$= 40ab\sqrt {ab} - 6ab\sqrt {ab} + 6ab\sqrt {ab} - 45ab\sqrt {ab} = - 5ab\sqrt {ab} .$$

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

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

a) $\sqrt {\frac{a}{b}} + \sqrt {ab} + \frac{a}{b}\sqrt {\frac{b}{a}} = \sqrt {\frac{{ab}}{{{b^2}}}} + \sqrt {ab} + \frac{a}{b}\sqrt {\frac{{ab}}{{{a^2}}}} = \frac{1}{b}\sqrt {ab} + \sqrt {ab} + \frac{1}{b}\sqrt {ab} = \left( {\frac{2}{b} + 1} \right)\sqrt {ab}$.

b) $\sqrt {\frac{m}{{1 - 2x + {x^2}}}} .\sqrt {\frac{{4m - 8mx + 4m{x^2}}}{{81}}} = \sqrt {\frac{m}{{{{\left( {1 - x} \right)}^2}}}.\frac{{4m{{\left( {1 - x} \right)}^2}}}{{81}}} = \sqrt {\frac{{4{m^2}}}{{81}}} = \frac{{\sqrt {4{m^2}} }}{{\sqrt {81} }} = \frac{{2m}}{9}.$ ( với $m > 0$ và $x \ne 1$)

Ta có: $\frac{1}{{a - \sqrt a }} + \frac{1}{{\sqrt a - 1}} = \frac{1}{{\sqrt a \left( {\sqrt a - 1} \right)}} + \frac{{\sqrt a }}{{\sqrt a \left( {\sqrt a - 1} \right)}} = \frac{{1 + \sqrt a }}{{\sqrt a \left( {\sqrt a - 1} \right)}}$

Do đó:$M = \frac{{1 + \sqrt a }}{{\sqrt a \left( {\sqrt a - 1} \right)}}:\frac{{\sqrt a + 1}}{{a - 2\sqrt a + 1}} = \frac{{1 + \sqrt a }}{{\sqrt a \left( {\sqrt a - 1} \right)}}.\frac{{{{\left( {\sqrt a - 1} \right)}^2}}}{{\sqrt a + 1}} = \frac{{\sqrt a - 1}}{{\sqrt a }} = 1 - \frac{1}{{\sqrt a }}.$

Vì $\sqrt a > 0$ nên $1 - \frac{1}{{\sqrt a }} < 1$ suy ra $M < 1.$

a) a) Ta có $x - 5\sqrt x + 6 = \left( {\sqrt x - 3} \right)\left( {\sqrt x - 2} \right)$

Điều kiện $\left\{ \begin{array}{l}x \ge 0\\\sqrt x \ne 3\\\sqrt x \ne 2\end{array} \right. \Leftrightarrow \left\{ \begin{array}{l}x \ge 0\\x \ne 9\\x \ne 4\end{array} \right.$

b) $B = \frac{{2\sqrt x - 9}}{{\left( {\sqrt x - 3} \right)\left( {\sqrt x - 2} \right)}} - \frac{{\sqrt x + 3}}{{\sqrt x - 2}} + \frac{{2\sqrt x + 1}}{{\sqrt x - 3}}$

$$= \frac{{2\sqrt x - 9 - \left( {\sqrt x + 3} \right)\left( {\sqrt x - 3} \right) + \left( {2\sqrt x + 1} \right)\left( {\sqrt x - 2} \right)}}{{\left( {\sqrt x - 3} \right)\left( {\sqrt x - 2} \right)}}$$

$= \frac{{2\sqrt x - 9 - x + 9 + 2x - 4\sqrt x + \sqrt x - 2}}{{\left( {\sqrt x - 3} \right)\left( {\sqrt x - 2} \right)}} = \frac{{x - \sqrt x - 2}}{{\left( {\sqrt x - 3} \right)\left( {\sqrt x - 2} \right)}}$

$$= \frac{{\left( {\sqrt x - 2} \right)\left( {\sqrt x + 1} \right)}}{{\left( {\sqrt x - 2} \right)\left( {\sqrt x - 3} \right)}} = \frac{{\sqrt x + 1}}{{\sqrt x - 3}}.$$

c) Ta có $B = \frac{{\sqrt x - 3 + 4}}{{\sqrt x - 3}} = 1 + \frac{4}{{\sqrt x - 3}}$

$$B > 1 \Leftrightarrow \frac{4}{{\sqrt x - 3}} > 0 \Leftrightarrow \sqrt x > 3 \Leftrightarrow x > 9$$

Vậy với $x > 9$ thì $B > 1.$

d) Vì $B = 1 + \frac{4}{{\sqrt x - 3}}$ nên $B \in Z \Leftrightarrow \sqrt x - 3$ là ước của 4. Do đó $\sqrt x - 3$ nhận các giá trị $\pm 1,{\rm{ }} \pm 2,{\rm{ }} \pm 4$.

Suy các giá trị thích hợp của x là $1,{\rm{ }}4,{\rm{ }}16,{\rm{ }}25,{\rm{ }}49$.