标题

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2019年中考数学一轮复习数与式模块-整式专题复习讲义-中考数学复习资料

2018年全国各省市整式专题中考真题汇编考点分析

知识点题量占比
幂的乘方与积的乘方48.00%
合并同类项1122.00%
多项式乘多项式24.00%
同底数幂的乘法816.00%
整式的加减36.00%
同底数幂的除法12.00%
算术平方根12.00%
完全平方公式24.00%
整式的混合运算48.00%
数学常识12.00%
单项式12.00%
平方差公式24.00%
整式的混合运算—化简求值816.00%
去括号与添括号12.00%
完全平方公式的几何背景12.00%

一.选择题(共31小题)

1.下列运算结果正确的是\((\)  \()\)

A.\(3a^{3}· 2a^{2}=6a^{6}\)                              

B.\({( – 2a)^2} =  – 4{a^2}\)         

C.\(\tan 45^\circ  = \frac{{\sqrt 2 }}{2}\)                   

D.\(\cos 30^\circ  = \frac{{\sqrt 3 }}{2}\)

\({\color{red}{【解答】}}\)解:\(A\)、原式\( = 6{a^5}\),故本选项错误;

\(B\)、原式\( = 4{a^2}\),故本选项错误;

\(C\)、原式\( = 1\),故本选项错误;

\(D\)、原式\( = \frac{{\sqrt 3 }}{2}\),故本选项正确.

故选:\(D\).

\({\color{red}{【总结】}}\)考查了同底数幂的乘法、幂的乘方与积的乘方、特殊角的三角函数值,属于基础计算题.

2.下列运算正确的是\((\)  \()\)

A.\({m^2} + 2{m^3} = 3{m^5}\)

B.\(m^{2}· m^{3}=m^{6}\)                                

C.\({( – m)^3} =  – {m^3}\)             

D.\({(mn)^3} = m{n^3}\)

\({\color{red}{【解答】}}\)解:\(A\)、\({m^2}\)与\(2{m^3}\)不是同类项,不能合并,此选项错误;

\(B\)、\(m^{2}· m^{3}=m^{5}\),此选项错误;

\(C\)、\({( – m)^3} =  – {m^3}\),此选项正确;

\(D\)、\({(mn)^3} = {m^3}{n^3}\),此选项错误;

故选:\(C\).

\({\color{red}{【总结】}}\)本题主要考查整式的运算,解题的关键是掌握合并同类项法则、同底数幂的乘法、幂的乘方与积的乘方.

3.计算\((a – 2)(a + 3)\)的结果是\((\)  \()\).

A.\({a^2} – 6\)  

B.\({a^2} + a – 6\)                                       

C.\({a^2} + 6\) 

D.\({a^2} – a + 6\)

\({\color{red}{【解答】}}\)解:\((a – 2)(a + 3) = {a^2} + a – 6\),

故选:\(B\).

\({\color{red}{【总结】}}\)此题考查多项式的乘法,关键是根据多项式乘法的法则解答.

4.计算\(a^{3}· (a^{3})^{2}\)的结果是\((\)  \()\)

A.\({a^8}\)           

B.\({a^9}\)            

C.\({a^{11}}\)   

D.\({a^{18}}\)

\({\color{red}{【解答】}}\)解:\(a^{3}· (a^{3})^{2}=a^{9}\),

故选:\(B\).

\({\color{red}{【总结】}}\)本题考查了幂的乘方,解决本题的关键是熟记幂的乘方公式.

5.下列计算正确的是\((\)  \()\)

A.\(a^{2}· a^{3}=a^{6}\)                                     

B.\({a^3} \div a = {a^3}\)                  

C.\(a – (b – a) = 2a – b\)                          

D.\({( – \frac{1}{2}a)^3} =  – \frac{1}{6}{a^3}\)

\({\color{red}{【解答】}}\)解:\(A\)、\(a^{2}· a^{3}=a^{5}\),故\(A\)错误;

\(B\)、\({a^3} \div a = {a^2}\),故\(B\)错误;

\(C\)、\(a – (b – a) = 2a – b\),故\(C\)正确;

\(D\)、\({( – \frac{1}{2}a)^3} =  – \frac{1}{8}{a^3}\),故\(D\)错误.

故选:\(C\).

\({\color{red}{【总结】}}\)本题考查合并同类项、积的乘方、同底数幂的乘除法,熟练掌握运算性质和法则是解题的关键.

6.计算\((a^{2})^{3}-5a^{3}· a^{3}\)的结果是\((\)  \()\)

A.\({a^5} – 5{a^6}\)                                 

B.\({a^6} – 5{a^9}\)                                 

C.\( – 4{a^6}\)   

D.\(4{a^6}\)

\({\color{red}{【解答】}}\)解:\((a^{2})^{3}-5a^{3}· a^{3}\)

\( = {a^6} – 5{a^6}\)

\( =  – 4{a^6}\).

故选:\(C\).

\({\color{red}{【总结】}}\)此题主要考查了幂的乘方运算、单项式乘以单项式,正确掌握运算法则是解题关键.

7.下列运算正确的是\((\)  \()\)

A.\({a^2} + {a^3} = {a^5}\)            

B.\({({a^2})^3} = {a^5}\)                  

C.\({a^4} – {a^3} = a\)                          

D.\({a^4} \div {a^3} = a\)

\({\color{red}{【解答】}}\)解:\(A\)、\({a^2}\)、\({a^3}\)不是同类项不能合并,故\(A\)错误;

\(B\)、\((a^{2})^{3}=a^{6}\),故\(B\)错误;

\(C\)、\({a^4}\)、\({a^3}\)不是同类项不能合并,故\(C\)错误;

\(D\)、\({a^4} \div {a^3} = a\),故..正确.

故选:\(D\).

\({\color{red}{【总结】}}\)本题考查合并同类项、幂的乘方、同底数幂的除法,熟练掌握运算性质和法则是解题的关键.

8.下列运算结果正确的是\((\)  \()\)

A.\(a^{2}· a^{3}=a^{6}\)                                     

B.\( – (a – b) =  – a + b\)                      

C.\({a^2} + {a^2} = 2{a^4}\)         

D.\({a^8} \div {a^4} = {a^2}\)

\({\color{red}{【解答】}}\)解:\(A\)、\(a^{2}· a^{3}=a^{5}\),故此选项错误;

\(B\)、\( – (a – b) =  – a + b\),正确;

\(C\)、\({a^2} + {a^2} = 2{a^2}\),故此选项错误;

\(D\)、\({a^8} \div {a^4} = {a^4}\),故此选项错误;

故选:\(B\).

\({\color{red}{【总结】}}\)此题主要考查了合并同类项以及同底数幂的乘除运算、去括号法则,正确掌握相关运算法则是解题关键.

9.下面是一位同学做的四道题:①\({(a + b)^2} = {a^2} + {b^2}\),②\({( – 2{a^2})^2} =  – 4{a^4}\),③\({a^5} \div {a^3} = {a^2}\),④\(a^{3}· a^{4}=a^{12}\).其中做对的一道题的序号是\((\)  \()\)

A.①                       

B.②                        

C.③                        

D.④

\({\color{red}{【解答】}}\)解:①\({(a + b)^2} = {a^2} + 2ab + {b^2}\),故此选项错误;

②\({( – 2{a^2})^2} = 4{a^4}\),故此选项错误;

③\({a^5} \div {a^3} = {a^2}\),正确;

④\(a^{3}· a^{4}=a^{7}\),故此选项错误.

故选:\(C\).

\({\color{red}{【总结】}}\)此题主要考查了完全平方公式以及同底数幂的乘除运算、积的乘方运算,正确掌握相关运算法则是解题关键.

10.下列计算错误的是\((\)  \()\)

A.\(a^{2}\div a^{0}· a^{2}=a^{4}\)                 

B.\(a^{2}\div (a^{0}· a^{2})=1\)                       

C.\({( – 1.5)^8} \div {( – 1.5)^7} =  – 1.5\)       

D.\( – {1.5^8} \div {( – 1.5)^7} =  – 1.5\)

\({\color{red}{【解答】}}\)解:∵\( a^{2}\div a^{0}· a^{2}=a^{4}\),

∴\( \)选项\(A\)不符合题意;

∵\( a^{2}\div (a^{0}· a^{2})=1\),

∴\( \)选项\(B\)不符合题意;

∵\( (-1.5)^{8}\div (-1.5)^{7}=-1.5\),

∴\( \)选项\(C\)不符合题意;

∵\( -1.5^{8}\div (-1.5)^{7}=1.5\),

∴\( \)选项\(D\)符合题意.

故选:\(D\).

\({\color{red}{【总结】}}\)此题主要考查了同底数幂的除法法则,同底数幂的乘法的运算方法,以及零指数幂的运算方法,同底数幂相除,底数不变,指数相减,要熟练掌握,解答此题的关键是要明确:①底数\(a \ne 0\),因为0不能做除数;②单独的一个字母,其指数是1,而不是0;③应用同底数幂除法的法则时,底数\(a\)可是单项式,也可以是多项式,但必须明确底数是什么,指数是什么.

11.已知\({5^x} = 3\),\({5^y} = 2\),则\({5^{2x – 3y}} = (\)  \()\)

A.\(\frac{3}{4}\)                                           

B.1                                  

C.\(\frac{2}{3}\)                                            

D.\(\frac{9}{8}\)

\({\color{red}{【解答】}}\)解:∵\( 5^{x}=3\),\({5^y} = 2\),

∴\( {5^{2x}} = {3^2} = 9\),\({5^{3y}} = {2^3} = 8\),

∴\( {5^{2x – 3y}} = \frac{{{5^{2x}}}}{{{5^{3y}}}} = \frac{9}{8}\).

故选:\(D\).

\({\color{red}{【总结】}}\)此题主要考查了同底数幂的除法法则,以及幂的乘方与积的乘方,同底数幂相除,底数不变,指数相减,要熟练掌握,解答此题的关键是要明确:①底数\(a \ne 0\),因为0不能做除数;②单独的一个字母,其指数是1,而不是0;③应用同底数幂除法的法则时,底数\(a\)可是单项式,也可以是多项式,但必须明确底数是什么,指数是什么.

12.计算\({( – a)^3} \div a\)结果正确的是\((\)  \()\)

A.\({a^2}\)           

B.\( – {a^2}\)      

C.\( – {a^3}\)      

D.\( – {a^4}\)

\({\color{red}{【解答】}}\)解:\({( – a)^3} \div a =  – {a^3} \div a =  – {a^{3 – 1}} =  – {a^2}\),

故选:\(B\).

\({\color{red}{【总结】}}\)此题主要考查了幂的乘方运算以及同底数幂的除法运算,正确掌握运算法则是解题关键.

13.下列计算正确的是\((\)  \()\)

A.\({(x + y)^2} = {x^2} + {y^2}\)  

B.\({( – \frac{1}{2}x{y^2})^3} =  – \frac{1}{6}{x^3}{y^6}\)  

C.\({x^6} \div {x^3} = {x^2}\)      

D.\(\sqrt {{{( – 2)}^2}}  = 2\)

\({\color{red}{【解答】}}\)解:\({(x + y)^2} = {x^2} + 2xy + {y^2}\),\(A\)错误;

\({( – \frac{1}{2}x{y^2})^3} =  – \frac{1}{8}{x^3}{y^6}\),\(B\)错误;

\({x^6} \div {x^3} = {x^3}\),\(C\)错误;

\(\sqrt {{{( – 2)}^2}}  = \sqrt 4  = 2\),\(D\)正确;

故选:\(D\).

\({\color{red}{【总结】}}\)本题考查的是完全平方公式、积的乘方、同底数幂的除法以及算术平方根的计算,掌握完全平方公式、积的乘方法则、同底数幂的除法法则和算术平方根的定义是解题的关键.

14.将\({9.5^2}\)变形正确的是\((\)  \()\)

A.\({9.5^2} = {9^2} + {0.5^2}\)  

B.\({9.5^2} = (10 + 0.5)(10 – 0.5)\)         

C.\({9.5^2} = {10^2} – 2 \times 10 \times 0.5 + {0.5^2}\)     

D.\({9.5^2} = {9^2} + 9 \times 0.5 + {0.5^2}\)

\({\color{red}{【解答】}}\)解:\({9.5^2} = {(10 – 0.5)^2} = {10^2} – 2 \times 10 \times 0.5 + {0.5^2}\),

故选:\(C\).

\({\color{red}{【总结】}}\)本题考查的是完全平方公式,完全平方公式:\({(a \pm b)^2} = {a^2} \pm 2ab + {b^2}\).可巧记为:“首平方,末平方,首末两倍中间放”.

15.下列运算正确的是\((\)  \()\)

A.\({x^2} + {x^2} = {x^4}\)            

B.\(x^{3}· x^{2}=x^{6}\)                                    

C.\(2{x^4} \div {x^2} = 2{x^2}\)                                            

D.\({(3x)^2} = 6{x^2}\)

\({\color{red}{【解答】}}\)解:\(A\)、\({x^2} + {x^2} = 2{x^2}\),选项\(A\)错误;

\(B\)、\(x^{3}· x^{2}=x^{3+2}=x^{5}\),选项\(B\)错误;

\(C\)、\(2{x^4} \div {x^2} = 2{x^{4 – 2}} = 2{x^2}\),选项\(C\)正确;

\(D\)、\((3x)^{2}=3^{2}· x^{2}=9x^{2}\),选项\(D\)错误.

故选:\(C\).

\({\color{red}{【总结】}}\)本题考查了整式的混合运算,牢记整式混合运算的运算法则是解题的关键.

16.下列运算正确的是\((\)  \()\)

A.\(a^{2}· a^{3}=a^{6}\)                                     

B.\({a^3} \div {a^{ – 3}} = 1\)       

C.\({(a – b)^2} = {a^2} – ab + {b^2}\) 

D.\({( – {a^2})^3} =  – {a^6}\)

\({\color{red}{【解答】}}\)解:\(A\)、\(a^{2}· a^{3}=a^{5}\),此选项错误;

\(B\)、\({a^3} \div {a^{ – 3}} = {a^6}\),此选项错误;

\(C\)、\({(a – b)^2} = {a^2} – 2ab + {b^2}\),此选项错误;

\(D\)、\({( – {a^2})^3} =  – {a^6}\),此选项正确;

故选:\(D\).

\({\color{red}{【总结】}}\)本题主要考查幂的运算,解题的关键是掌握同底数幂的乘法、完全平方公式及同底数幂的除法、幂的乘方的运算法则.

17.下列计算正确的是\((\)  \()\)

A.\({a^4} + {a^5} = {a^9}\)           

B.\({(2{a^2}{b^3})^2} = 4{a^4}{b^6}\) 

C.\( – 2a(a + 3) =  – 2{a^2} + 6a\)       

D.\({(2a – b)^2} = 4{a^2} – {b^2}\)

\({\color{red}{【解答】}}\)解:\(A\)、\({a^4}\)与\({a^5}\)不是同类项,不能合并,故本选项错误;

\(B\)、\({(2{a^2}{b^3})^2} = 4{a^4}{b^6}\),故本选项正确;

\(C\)、\( – 2a(a + 3) =  – 2{a^2} – 6a\),故本选项错误;

\(D\)、\({(2a – b)^2} = 4{a^2} – 4ab + {b^2}\),故本选项错误;

故选:\(B\).

\({\color{red}{【总结】}}\)本题主要考查了合并同类项的法则、幂的乘方与积的乘方、单项式乘多项式法则以及完全平方公式,熟练掌握运算法则是解题的关键.

18.下列计算正确的是\((\)  \()\)

A.\(2x + 3y = 5xy\)                                   

B.\({( – 2{x^2})^3} =  – 6{x^6}\)

C.\(3y^{2}· (-y)=-3y^{2}\)                                

D.\(6{y^2} \div 2y = 3y\)

\({\color{red}{【解答】}}\)解:A、原式\( = 2x + 3y\),故\(A\)错误;

B、原式\( =  – 8{x^6}\),故\(B\)错误;

C、原式\( =  – 3{y^3}\),故\(C\)错误;

故选:\(D\).

\({\color{red}{【总结】}}\)本题考查整式的运算法则,解题的关键是熟练运用整式的运算法则,本题属于基础题型.

19.下列运算正确的是\((\)  \()\)

A.\({a^2} + {a^2} = {a^4}\)            

B.\({a^3} \div a = {a^3}\)                  

C.\(a^{2}· a^{3}=a^{5}\)                                     

D.\({({a^2})^4} = {a^6}\)

\({\color{red}{【解答】}}\)解:\(A\)、\({a^2} + {a^2} = 2{a^2}\),故\(A\)错误;

\(B\)、\({a^3} \div a = {a^2}\),故\(B\)错误;

\(C\)、\(a^{2}· a^{3}=a^{5}\),故\(C\)正确;

\(D\)、\({({a^2})^3} = {a^8}\),故\(D\)错误.

故选:\(C\).

\({\color{red}{【总结】}}\)本题考查合并同类项、同底数幂的乘法、幂的乘方、同底数幂的除法,熟练掌握运算性质和法则是解题的关键.

20.下列运算正确的是\((\)  \()\)

A.\(a^{2}· a^{5}=a^{10}\)                                   

B.\({(3{a^3})^2} = 6{a^6}\)           

C.\({(a + b)^2} = {a^2} + {b^2}\)  

D.\((a + 2)(a – 3) = {a^2} – a – 6\)

\({\color{red}{【解答】}}\)解:\(A\)、原式\( = {a^7}\),不符合题意;

\(B\)、原式\( = 9{a^6}\),不符合题意;

\(C\)、原式\( = {a^2} + 2ab + {b^2}\),不符合题意;

\(D\)、原式\( = {a^2} – a – 6\),符合题意,

故选:\(D\).

\({\color{red}{【总结】}}\)此题考查了整式的混合运算,熟练掌握运算法则是解本题的关键.

21.下列运算结果正确的是\((\)  \()\)

A.\(a^{3}· a^{2}=a^{5}\)                                     

B.\({({a^3})^2} = {a^5}\)                  

C.\({a^3} + {a^2} = {a^5}\)            

D.\({a^{ – 2}} =  – {a^2}\)

\({\color{red}{【解答】}}\)解:\(A\)、\(a^{3}· a^{2}=a^{5}\),正确,故本选项符合题意;

\(B\)、\({({a^3})^2} = {a^6}\),故本选项不符合题意;

\(C\)、不是同类项不能合并,故本选项不符合题意;

\(D\)、\({a^{ – 2}} = \frac{1}{{{a^2}}}\),故本选项不符合题意,

故选:\(A\).

\({\color{red}{【总结】}}\)本题考查积的乘方,幂的乘方,负指数幂的定义,解题的关键是熟练掌握基本知识,属于中考常考题型.

22.下列运算正确的是\((\)  \()\)

A.\({({a^2})^3} = {a^5}\)                  

B.\(a^{4}· a^{2}=a^{8}\)                                     

C.\({a^6} \div {a^3} = {a^2}\)      

D.\({(ab)^3} = {a^3}{b^3}\)

\({\color{red}{【解答】}}\)解:∵\( (a^{2})^{3}=a^{6}\),

∴\( \)选项\(A\)不符合题意;

∵\( a^{4}· a^{2}=a^{6}\),

∴\( \)选项\(B\)不符合题意;

∵\( a^{6}\div a^{3}=a^{3}\),

∴\( \)选项\(C\)不符合题意;

∵\( (ab)^{3}=a^{3}b^{3}\),

∴\( \)选项\(D\)符合题意.

故选:\(D\).

\({\color{red}{【总结】}}\)此题主要考查了同底数幂的除法法则,同底数幂的乘法的运算方法,以及幂的乘方与积的乘方的运算方法,同底数幂相除,底数不变,指数相减,要熟练掌握,解答此题的关键是要明确:①底数\(a \ne 0\),因为0不能做除数;②单独的一个字母,其指数是1,而不是0;③应用同底数幂除法的法则时,底数\(a\)可是单项式,也可以是多项式,但必须明确底数是什么,指数是什么.

23.下列运算正确的是\((\)  \()\)

A.\(x – 2x =  – x\)                                       

B.\(2x – y =  – xy\)                                    

C.\({x^2} + {x^2} = {x^4}\)            

D.\({(x – 1)^2} = {x^2} – 1\)

\({\color{red}{【解答】}}\)解:B、原式\( = 2x – y\),故\(B\)错误;

C、原式\( = 2{x^2}\),故\(C\)错误;

D、原式\( = {x^2} – 2x + 1\),故\(D\)错误;

故选:\(A\).

\({\color{red}{【总结】}}\)本题考查整式的运算法则,解题的关键是熟练运用整式的运算法则,本题属于基础题型.

24.下列计算正确的是\((\)  \()\)

A.\({a^3} + {a^3} = 2{a^3}\)         

B.\(a^{3}· a^{2}=a^{6}\)                                     

C.\({a^6} \div {a^2} = {a^3}\)      

D.\({({a^3})^2} = {a^5}\)

\({\color{red}{【解答】}}\)解:∵\( a^{3}+a^{3}=2a^{3}\),

∴\( \)选项\(A\)符合题意;

∵\( a^{3}· a^{2}=a^{5}\),

∴\( \)选项\(B\)不符合题意;

∵\( a^{6}\div a^{2}=a^{4}\),

∴\( \)选项\(C\)不符合题意;

∵\( (a^{3})^{2}=a^{6}\),

∴\( \)选项\(D\)不符合题意.

故选:\(A\).

\({\color{red}{【总结】}}\)此题主要考查了同底数幂的除法法则,同底数幂的乘法的运算方法,合并同类项的方法,以及幂的乘方与积的乘方的运算方法,解答此题的关键是要明确:①底数\(a \ne 0\),因为0不能做除数;②单独的一个字母,其指数是1,而不是0;③应用同底数幂除法的法则时,底数\(a\)可是单项式,也可以是多项式,但必须明确底数是什么,指数是什么.

25.下列运算正确的是\((\)  \()\)

A.\({a^2} + {a^2} = 2{a^4}\)         

B.\({a^6} \div {a^2} = {a^3}\)      

C.\({( – {a^3})^2} = {a^6}\)            

D.\({(ab)^2} = a{b^2}\)

\({\color{red}{【解答】}}\)解:\(A\)、\({a^2} + {a^2} = 2{a^2}\),故\(A\)错误;

\(B\)、\({a^6} \div {a^2} = {a^4}\),故\(B\)错误;

\(C\)、\({( – {a^3})^2} = {a^6}\),故\(C\)正确;

\(D\)、\({(ab)^2} = {a^2}{b^2}\),故\(D\)错误.

故选:\(C\).

\({\color{red}{【总结】}}\)本题考查合并同类项、同底数幂的除法、积的乘方,熟练掌握运算性质和法则是解题的关键.

26.下列运算正确的是\((\)  \()\)

A.\({a^8} \div {a^4} = {a^2}\)     

B.\({({a^2})^2} = {a^4}\)                  

C.\(a^{2}· a^{3}=a^{6}\)                                     

D.\({a^2} + {a^2} = 2{a^4}\)

\({\color{red}{【解答】}}\)解:\(A\)、\({a^8} \div {a^6} = {a^4}\),故此选项错误;

\(B\)、\({({a^2})^2} = {a^4}\),故原题计算正确;

\(C\)、\(a^{2}· a^{3}=a^{5}\),故此选项错误;

\(D\)、\({a^2} + {a^2} = 2{a^2}\),故此选项错误;

故选:\(B\).

\({\color{red}{【总结】}}\)此题主要考查了同底数幂的乘除运算以及幂的乘方运算和合并同类项,正确掌握相关运算法则是解题关键.

27.若\({2^n} + {2^n} + {2^n} + {2^n} = 2\),则\(n = (\)  \()\)

A.\( – 1\)                  

B.\( – 2\)                   

C.0                                  

D.\(\frac{1}{4}\)

\({\color{red}{【解答】}}\)解:∵\( 2^{n}+2^{n}+2^{n}+2^{n}=2\),

∴\( 4· 2^{n}=2\),

∴\( 2· 2^{n}=1\),

∴\( {2^{1 + n}} = 1\),

∴\( 1 + n = 0\),

∴\( n =  – 1\).

故选:\(A\).

\({\color{red}{【总结】}}\)本题考查了同底数幂的乘法:同底数幂相乘,底数不变,指数相加,即\(a^{m}· a^{n}=a\)\(^{m + n}(m\),\(n\)是正整数).

28.在矩形\(ABCD\)内,将两张边长分别为\(a\)和\(b(a > b)\)的正方形纸片按图1,图2两种方式放置(图1,图2中两张正方形纸片均有部分重叠),矩形中未被这两张正方形纸片覆盖的部分用阴影表示,设图1中阴影部分的面积为\({S_1}\),图2中阴影部分的面积为\({S_2}\).当\(AD – AB = 2\)时,\({S_2} – {S_1}\)的值为\((\)  \()\)

A.\(2a\)                     

B.\(2b\)                     

C.\(2a – 2b\)         

D.\( – 2b\)

\({\color{red}{【解答】}}\)解:\(S_{1}=(AB-a)· a+(CD-b)(AD-a)=(AB-a)· a+(AB-b)(AD-a)\),

\({S_2} = AB(AD – a) + (a – b)(AB – a)\),

∴\( {{S}_{2}}-{{S}_{1}}=AB(AD-a)+(a-b)(AB-a)-(AB-a)· a-(AB-b)(AD-a)\)

\( = (AD – a)(AB – AB + b) + (AB – a)(a – b – a)\)

\(=b· AD-ab-b· AB+ab=b(AD-AB)\)

\( = 2b\)

故选:\(B\).

\({\color{red}{【总结】}}\)本题考查了整式的混合运算:整体”思想在整式运算中较为常见,适时采用整体思想可使问题简单化,并且迅速地解决相关问题,此时应注意被看做整体的代数式通常要用括号括起来.也考查了正方形的性质.

29.下面运算结果为\({a^6}\)的是\((\)  \()\)

A.\({a^3} + {a^3}\)                                   

B.\({a^8} \div {a^2}\)                             

C.\(a^{2}· a^{3}\)     

D.\({( – {a^2})^3}\)

\({\color{red}{【解答】}}\)解:\(A\)、\({a^3} + {a^3} = 2{a^3}\),此选项不符合题意;

\(B\)、\({a^8} \div {a^2} = {a^6}\),此选项符合题意;

\(C\)、\(a^{2}· a^{3}=a^{5}\),此选项不符合题意;

\(D\)、\({( – {a^2})^3} =  – {a^6}\),此选项不符合题意;

故选:\(B\).

\({\color{red}{【总结】}}\)本题主要考查整式的运算,解题的关键是掌握合并同类项法则、同底数幂的除法、同底数幂的乘法及幂的乘方.

30.我国南宋数学家杨辉所著的《详解九章算术》一书中,用如图的三角形解释二项式\({(a + b)^n}\)的展开式的各项系数,此三角形称为“杨辉三角”

根据”杨辉三角”请计算\({(a + b)^8}\)的展开式中从左起第四项的系数为\((\)  \()\)

A.84                             

B.56                             

C.35                             

D.28

\({\color{red}{【解答】}}\)解:找规律发现\({(a + b)^4}\)的第四项系数为\(4 = 3 + 1\);

\({(a + b)^5}\)的第四项系数为\(10 = 6 + 4\);

\({(a + b)^6}\)的第四项系数为\(20 = 10 + 10\);

\({(a + b)^7}\)的第四项系数为\(35 = 15 + 20\);

∴\( {(a + b)^8}\)第四项系数为\(21 + 35 = 56\).

故选:\(B\).

\({\color{red}{【总结】}}\)此题考查了数字变化规律,通过观察、分析、归纳发现其中的规律,并应用发现的规律解决问题的能力.

31.下列计算正确的是\((\)  \()\)

A.\(a^{3}· a^{3}=2a^{3}\)                                   

B.\({a^2} + {a^2} = {a^4}\)            

C.\({a^6} \div {a^2} = {a^3}\)      

D.\({( – 2{a^2})^3} =  – 8{a^6}\)

\({\color{red}{【解答】}}\)解:\(A\)、\(a^{3}· a^{3}=a^{6}\),此选项错误;

\(B\)、\({a^2} + {a^2} = 2{a^2}\),此选项错误;

\(C\)、\({a^6} \div {a^2} = {a^4}\),此选项错误;

\(D\)、\({( – 2{a^2})^3} =  – 8{a^6}\),此选项正确;

故选:\(D\).

\({\color{red}{【总结】}}\)本题主要考查幂的运算,解题的关键是掌握同底数幂的乘法、合并同类项法则及同底数幂的除法、积的乘方与幂的乘方运算法则.



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