Bài tập cuối chương 3 lớp 9 (có lời giải)

a) Ta có \(\sqrt {\frac{9}{{16}}:\frac{{25}}{{36}}} - \sqrt {\frac{{49}}{8}} :\sqrt {3\frac{1}{8}} = \sqrt {\frac{9}{{16}}:\frac{{25}}{{36}}} - \sqrt {\frac{{49}}{8}:\frac{{25}}{8}} = \frac{3}{4}:\frac{5}{6} - \sqrt {\frac{{49}}{{25}}} = \frac{9}{{10}} - \frac{7}{5} = - \frac{1}{2}.\)

b) \(\sqrt {{{45,8}^2} - {{44,2}^2}} - \sqrt {6\left[ {{{(\sqrt 2 + 1)}^2} + {{(\sqrt 2 - 1)}^2}} \right]} = \sqrt {1,6.90} - \sqrt {6.6} = 4.3 - 6 = 6.\)

Lời giải

\(\begin{array}{l}a)\;\frac{1}{{34}}\sqrt {\frac{{{{165}^2} - {{124}^2}}}{{164}}} + 4\sqrt {\frac{{32}}{{{{176}^2} - {{112}^2}}}} = \frac{1}{{34}}\sqrt {\frac{{41.289}}{{164}}} + 4\sqrt {\frac{{32}}{{64.288}}} = \frac{1}{{34}}\sqrt {\frac{{289}}{4}} + 4\sqrt {\frac{1}{{576}}} \\ = \frac{1}{{34}} \cdot \frac{{17}}{2} + 4 \cdot \frac{1}{{24}} = \frac{1}{4} + \frac{1}{6} = \frac{5}{{12}}\end{array}\)

b) \(\frac{{5\left( {\sqrt 6 - 1} \right)}}{{\sqrt 6 + 1}} + \frac{{\sqrt 2 - \sqrt 3 }}{{\sqrt 2 + \sqrt 3 }} = \frac{{5{{(\sqrt 6 - 1)}^2}}}{{6 - 1}} + \frac{{{{(\sqrt 2 - \sqrt 3 )}^2}}}{{2 - 3}} = 7 - 2\sqrt 6 - 5 + 2\sqrt 6 = 2.\)

a) Ta có \(\frac{{3 + 3\sqrt 5 - \sqrt 2 - \sqrt {10} }}{{6 + 2\sqrt 5 }} = \frac{{3\left( {1 + \sqrt 5 } \right) - \sqrt 2 \left( {1 + \sqrt 5 } \right)}}{{{{(1 + \sqrt 5 )}^2}}} = \frac{{\left( {1 + \sqrt 5 } \right)\left( {3 - \sqrt 2 } \right)}}{{{{(1 + \sqrt 5 )}^2}}} = \frac{{3 - \sqrt 2 }}{{1 + \sqrt 5 }}.\)
b) \(\frac{{x\sqrt x + x\sqrt y - y\sqrt x - y\sqrt y }}{{x - y + y\sqrt x - y\sqrt y }} = \frac{{x\left( {\sqrt x + \sqrt y } \right) - y\left( {\sqrt x + \sqrt y } \right)}}{{\left( {\sqrt x - \sqrt y } \right)\left( {\sqrt x + \sqrt y } \right) + y\left( {\sqrt x - \sqrt y } \right)}} = \frac{{\left( {\sqrt x + \sqrt y } \right)\left( {x - y} \right)}}{{\left( {\sqrt x - \sqrt y } \right)\left( {\sqrt x + \sqrt y + y} \right)}}\)

Điều kiện : \(x \ge 0;x \ne 4;x \ne 9\). Khi đó ta có

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

\( = \frac{{2\sqrt x - 9 - x + 9 + 2x - 4\sqrt x + \sqrt x - 2}}{{\left( {\sqrt x - 2} \right)\left( {\sqrt x - 3} \right)}} = \frac{{x - \sqrt x - 2}}{{\left( {\sqrt x - 2} \right)\left( {\sqrt x - 3} \right)}} = \frac{{\left( {\sqrt x + 1} \right)\left( {\sqrt x - 2} \right)}}{{\left( {\sqrt x - 2} \right)\left( {\sqrt x - 3} \right)}} = \frac{{\sqrt x + 1}}{{\sqrt x - 3}}\)

a) Điều kiện : \(x \ge 0;x \ne 9\). Khi đó ta có

\(P = \frac{{2\sqrt x \left( {\sqrt x  - 3} \right) + \left( {\sqrt x  + 1} \right)\left( {\sqrt x  + 3} \right) - 3 + 11\sqrt x }}{{\left( {\sqrt x  + 3} \right)\left( {\sqrt x  - 3} \right)}} = \frac{{2x - 6\sqrt x  + x + 4\sqrt x  + 3 - 3 + 11\sqrt x }}{{\left( {\sqrt x  + 3} \right)\left( {\sqrt x  - 3} \right)}}\)

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

b) Ta có \(x = \frac{{7 + 4\sqrt 3 }}{4} = {\left( {\frac{{2 + \sqrt 3 }}{2}} \right)^2} \Rightarrow \sqrt x  = \frac{{2 + \sqrt 3 }}{2}\).
Do đó \({\rm{P}} = \frac{{3 \cdot \frac{{2 + \sqrt 3 }}{2}}}{{\frac{{2 + \sqrt 3 }}{2} - 3}} = \frac{{6 + 3\sqrt 3 }}{2} \cdot \frac{2}{{\sqrt 3  - 4}}\)\( = \frac{{6 + 3\sqrt 3 }}{{\sqrt 3  - 4}} = \frac{{\left( {6 + 3\sqrt 3 } \right)\left( {\sqrt 3  + 4} \right)}}{{\left( {\sqrt 3  - 4} \right)\left( {\sqrt 3  + 4} \right)}}\; = \frac{{6\sqrt 3  + 24 + 9 + 12\sqrt 3 }}{{3 - 16}} = \frac{{ - \left( {33 + 18\sqrt 3 } \right)}}{{13}}\begin{array}{*{20}{r}}{}&\;\\{}&.\end{array}\)

a) Điều kiện : \(x \ge 0;x \ne 9\). Khi đó ta có

\(P = \frac{{\left( {\sqrt x  - 3} \right) + 5\left( {\sqrt x  + 3} \right) + 6}}{{\left( {\sqrt x  + 3} \right)\left( {\sqrt x  - 3} \right)}} \cdot \frac{{\sqrt x  + 2}}{6} = \frac{{\sqrt x  - 3 + 5\sqrt x  + 15 + 6}}{{\left( {\sqrt x  + 3} \right)\left( {\sqrt x  - 3} \right)}} \cdot \frac{{\sqrt x  + 2}}{6} = \frac{{6\sqrt x  + 18}}{{\left( {\sqrt x  + 3} \right)\left( {\sqrt x  - 3} \right)}} \cdot \frac{{\sqrt x  + 2}}{6}\)

\( = \frac{{6\left( {\sqrt x  + 3} \right)}}{{\left( {\sqrt x  + 3} \right)\left( {\sqrt x  - 3} \right)}} \cdot \frac{{\sqrt x  + 2}}{6} = \frac{{\sqrt x  + 2}}{{\sqrt x  - 3}}\)

Ta có \(P = \frac{{\sqrt x  + 2}}{{\sqrt x  - 3}} = \frac{{\sqrt x  - 3 + 5}}{{\sqrt x  - 3}} = 1 + \frac{5}{{\sqrt x  - 3}}\).
P có giá trị nguyên \( \Leftrightarrow \frac{5}{{\sqrt x  - 3}}\) có giá trị nguyên\( \Leftrightarrow \sqrt x  - 3 \in U\left( 5 \right) \Leftrightarrow \sqrt x  - 3 \in \left\{ { \pm 1; \pm 5} \right\}\)

Ta có bảng sau :

Điều kiện : \(x,y > 0;x \ne y\). Khi đó ta có

\(P = \frac{2}{{\sqrt {xy} }}:{\left( {\frac{{\sqrt y  - \sqrt x }}{{\sqrt {xy} }}} \right)^2} - \frac{{x + y}}{{{{(\sqrt x  - \sqrt y )}^2}}}\)\( = \frac{2}{{\sqrt {xy} }} \cdot \frac{{{{(\sqrt {xy} )}^2}}}{{{{(\sqrt x  - \sqrt y )}^2}}} - \frac{{x + y}}{{{{(\sqrt x  - \sqrt y )}^2}}}\)

\( = \frac{{2\sqrt {xy} }}{{{{(\sqrt x  - \sqrt y )}^2}}} - \frac{{x + y}}{{{{(\sqrt x  - \sqrt y )}^2}}}\frac{{ - \left( {x - 2\sqrt {xy}  + y} \right)}}{{{{(\sqrt x  - \sqrt y )}^2}}} =  - \frac{{{{(\sqrt x  - \sqrt y )}^2}}}{{{{(\sqrt x  - \sqrt y )}^2}}} =  - 1\)

 Vậy giá trị của của \(P\) không phụ thuợc vào \(x\) và \(y\).

a) Điều kiện : \(x > 0;x \ne 9\). Khi đó ta có

\(P = \frac{{x + 3 + \sqrt x  - 3}}{{\left( {\sqrt x  - 3} \right)\left( {\sqrt x  + 3} \right)}} \cdot \frac{{\sqrt x  - 3}}{{\sqrt x }} = \frac{{\sqrt x \left( {\sqrt x  + 1} \right)}}{{\left( {\sqrt x  - 3} \right)\left( {\sqrt x  + 3} \right)}} \cdot \frac{{\sqrt x  - 3}}{{\sqrt x }} = \frac{{\sqrt x  + 1}}{{\sqrt x  + 3}}\)
b) Xét hiệu\(P - \frac{1}{3} = \frac{{\sqrt x  + 1}}{{\sqrt x  + 3}} - \frac{1}{3} = \frac{{3\sqrt x  + 3 - \sqrt x  - 3}}{{3\left( {\sqrt x  + 3} \right)}} = \frac{{2\sqrt x }}{{3\left( {\sqrt x  + 3} \right)}} > 0({\rm{\;v\`i \;}}x > 0){\rm{.\;}}\)

a) Điều kiện : \(x > 0\). Khi đó ta có \(P = \frac{{\left( {x - \sqrt x  + 1} \right) - \left( {x + 2} \right)}}{{\left( {\sqrt x  + 1} \right)\left( {x - \sqrt x  + 1} \right)}}:\frac{2}{{\sqrt x }} = \frac{{ - \left( {\sqrt x  + 1} \right)}}{{\left( {\sqrt x  + 1} \right)\left( {x - \sqrt x  + 1} \right)}} \cdot \frac{{\sqrt x }}{2} = \frac{{ - \sqrt x }}{{2\left( {x - \sqrt x  + 1} \right)}}\)