2025年化简计算题库及答案

2025年化简计算题库及答案一、整式化简1.化简：3(2x²5xy+y²)2(3x²4xy2y²)解答：原式=6x²15xy+3y²6x²+8xy+4y²=(-15xy+8xy)+(3y²+4y²)=-7xy+7y²2.化简：(2a+3b)(ab)(a2b)²解答：展开得2a²2ab+3ab3b²(a²4ab+4b²)=2a²+ab3b²a²+4ab4b²=a²+5ab7b²3.化简：(x+2y)(x2y)+(x3y)²2x(xy)解答：平方差展开得x²4y²，完全平方展开得x²6xy+9y²，最后项展开得2x²2xy，合并后x²4y²+x²6xy+9y²2x²+2xy=(-6xy+2xy)+(-4y²+9y²)=-4xy+5y²4.化简：5(3m²2mn+n²)(2m²+mn4n²)3(4m²5mn+2n²)解答：展开得15m²10mn+5n²2m²mn+4n²12m²+15mn6n²=(15m²-2m²-12m²)+(-10mn-mn+15mn)+(5n²+4n²-6n²)=m²+4mn+3n²5.化简：(3p2q)³(3p+2q)(9p²4q²)解答：立方展开得27p³54p²q+36pq²8q³，后项为(3p+2q)(3p-2q)(3p+2q)=(9p²-4q²)(3p+2q)=27p³+18p²q12pq²8q³，相减得27p³54p²q+36pq²8q³27p³18p²q+12pq²+8q³=(-54p²q-18p²q)+(36pq²+12pq²)=-72p²q+48pq²二、分式化简1.化简：[(a²9)/(a²+6a+9)]÷[(a3)/(a+3)]+2/(a+3)解答：因式分解得[(a-3)(a+3)/(a+3)²]×[(a+3)/(a-3)]+2/(a+3)=1+2/(a+3)=(a+3+2)/(a+3)=(a+5)/(a+3)2.化简：[1/(x1)1/(x+1)]÷[2/(x²1)]解答：通分前项得[(x+1)-(x-1)]/[(x-1)(x+1)]=2/(x²-1)，除以2/(x²-1)得2/(x²-1)×(x²-1)/2=13.化简：[(x²4x+4)/(x²4)]×[(x+2)/(x²2x)]1/x解答：因式分解得[(x-2)²/(x-2)(x+2)]×[(x+2)/x(x-2)]1/x=(x-2)/(x+2)×(x+2)/[x(x-2)]1/x=1/x1/x=04.化简：[(y+2)/(y2)(y2)/(y+2)]÷[(4y)/(y²4)]解答：前项通分[(y+2)²-(y-2)²]/[(y-2)(y+2)]=[y²+4y+4-y²+4y-4]/(y²-4)=8y/(y²-4)，除以4y/(y²-4)得8y/(y²-4)×(y²-4)/4y=25.化简：[(a1)/(a²+a)+(a+1)/(a²a)]÷(a²+1)/(a²1)解答：因式分解分母得[(a-1)/a(a+1)+(a+1)/a(a-1)]×(a²-1)/(a²+1)，通分前项得[(a-1)²+(a+1)²]/[a(a+1)(a-1)]×(a-1)(a+1)/(a²+1)=[a²-2a+1+a²+2a+1]/[a(a²-1)]×(a²-1)/(a²+1)=2(a²+1)/[a(a²-1)]×(a²-1)/(a²+1)=2/a三、根式化简1.化简：√(48)+3√(12)5√(27)+√(75)解答：化简为4√3+3×2√35×3√3+5√3=4√3+6√3-15√3+5√3=02.化简：(√(18)√(8))×√(2)+√(32)÷√(2)解答：计算得(3√22√2)×√2+4√2÷√2=√2×√2+4=2+4=63.化简：(√(5)+2)/(√(5)2)(√(5)2)/(√(5)+2)解答：有理化分母得[(√5+2)²-(√5-2)²]/[(√5-2)(√5+2)]=[(5+4√5+4)-(5-4√5+4)]/(5-4)=(9+4√5-9+4√5)/1=8√54.化简：√(x³y)x√(xy)+√(4xy³)（x>0，y>0）解答：化简为x√(xy)x√(xy)+2y√(xy)=2y√(xy)5.化简：(√(3)+1)²√(12)+(√(3)1)⁻¹解答：展开得3+2√3+12√3+1/(√3-1)=4+(√3+1)/[(√3-1)(√3+1)]=4+(√3+1)/2=(8+√3+1)/2=(9+√3)/2四、绝对值化简1.化简：|x5|+|x+3|（分情况讨论）解答：关键点x=5和x=-3，①x≥5时，原式=(x-5)+(x+3)=2x-2；②-3≤x<5时，原式=(5-x)+(x+3)=8；③x<-3时，原式=(5-x)+(-x-3)=2-2x2.化简：|2a1||a+2|（a为实数，分情况讨论）解答：关键点a=1/2和a=-2，①a≥1/2时，原式=(2a-1)-(a+2)=a-3；②-2≤a<1/2时，原式=(1-2a)-(a+2)=-3a-1；③a<-2时，原式=(1-2a)-(-a-2)=-a+33.化简：|m4|2|m+1|（当m≥4时；当-1≤m<4时；当m<-1时）解答：①m≥4时，原式=(m-4)-2(m+1)=-m-6；②-1≤m<4时，原式=(4-m)-2(m+1)=4-m-2m-2=2-3m；③m<-1时，原式=(4-m)-2(-m-1)=4-m+2m+2=m+64.化简：|3t|+|t5|（t为实数，求最小值及对应t值）解答：当3≤t≤5时，原式=(t-3)+(5-t)=2，为最小值；t<3时，原式=(3-t)+(5-t)=8-2t>2；t>5时，原式=(t-3)+(t-5)=2t-8>2，故最小值为2，t∈[3,5]5.化简：|x²4||x2|（x>1，分情况讨论）解答：x²-4=(x-2)(x+2)，关键点x=2（x>1），①1<x<2时，原式=(4-x²)-(2-x)=4-x²-2+x=-x²+x+2；②x≥2时，原式=(x²-4)-(x-2)=x²-x-2五、指数运算化简1.化简：(2a²b⁻³)³×(a⁻¹b²)⁻²÷(4a⁻²b⁻¹)解答：计算得8a⁶b⁻⁹×a²b⁻⁴÷4a⁻²b⁻¹=2a⁶⁺²⁺²b⁻⁹⁻⁴⁺¹=2a¹⁰b⁻¹²=2a¹⁰/b¹²2.化简：(3x⁻²y³)²÷(2x³y⁻⁴)⁻¹×(6x⁻¹y⁻²)解答：计算得9x⁻⁴y⁶÷(x⁻³y⁴/2)×6x⁻¹y⁻²=9x⁻⁴y⁶×2x³y⁴×6x⁻¹y⁻²=108x⁻⁴⁺³⁻¹y⁶⁺⁴⁻²=108x⁻²y⁸=108y⁸/x²3.化简：[(2⁻¹a³b⁻²)²×(4a⁻²b³)⁻¹]÷(8a⁻⁴b⁵)⁻²解答：计算得(2⁻²a⁶b⁻⁴×4⁻¹a²b⁻³)÷(8⁻²a⁸b⁻¹⁰)=(2⁻²×2⁻²a⁶⁺²b⁻⁴⁻³)÷(2⁻⁶a⁸b⁻¹⁰)=(2⁻⁴a⁸b⁻⁷)×2⁶a⁻⁸b¹⁰=2²b³=4b³4.化