2025年化簡(jiǎn)計(jì)算題庫(kù)及答案一、整式化簡(jiǎn)1.化簡(jiǎn)：3(2x25xy+y2)2(3x24xy2y2)解答：原式=6x215xy+3y26x2+8xy+4y2=(-15xy+8xy)+(3y2+4y2)=-7xy+7y22.化簡(jiǎn)：(2a+3b)(ab)(a2b)2解答：展開(kāi)得2a22ab+3ab3b2(a24ab+4b2)=2a2+ab3b2a2+4ab4b2=a2+5ab7b23.化簡(jiǎn)：(x+2y)(x2y)+(x3y)22x(xy)解答：平方差展開(kāi)得x24y2，完全平方展開(kāi)得x26xy+9y2，最后項(xiàng)展開(kāi)得2x22xy，合并后x24y2+x26xy+9y22x2+2xy=(-6xy+2xy)+(-4y2+9y2)=-4xy+5y24.化簡(jiǎn)：5(3m22mn+n2)(2m2+mn4n2)3(4m25mn+2n2)解答：展開(kāi)得15m210mn+5n22m2mn+4n212m2+15mn6n2=(15m2-2m2-12m2)+(-10mn-mn+15mn)+(5n2+4n2-6n2)=m2+4mn+3n25.化簡(jiǎn)：(3p2q)3(3p+2q)(9p24q2)解答：立方展開(kāi)得27p354p2q+36pq28q3，后項(xiàng)為(3p+2q)(3p-2q)(3p+2q)=(9p2-4q2)(3p+2q)=27p3+18p2q12pq28q3，相減得27p354p2q+36pq28q327p318p2q+12pq2+8q3=(-54p2q-18p2q)+(36pq2+12pq2)=-72p2q+48pq2二、分式化簡(jiǎn)1.化簡(jiǎn)：[(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.化簡(jiǎn)：[1/(x1)1/(x+1)]÷[2/(x21)]解答：通分前項(xiàng)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.化簡(jiǎn)：[(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.化簡(jiǎn)：[(y+2)/(y2)(y2)/(y+2)]÷[(4y)/(y24)]解答：前項(xiàng)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.化簡(jiǎn)：[(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)，通分前項(xiàng)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、根式化簡(jiǎn)1.化簡(jiǎn)：√(48)+3√(12)5√(27)+√(75)解答：化簡(jiǎn)為4√3+3×2√35×3√3+5√3=4√3+6√3-15√3+5√3=02.化簡(jiǎn)：(√(18)√(8))×√(2)+√(32)÷√(2)解答：計(jì)算得(3√22√2)×√2+4√2÷√2=√2×√2+4=2+4=63.化簡(jiǎn)：(√(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.化簡(jiǎn)：√(x3y)x√(xy)+√(4xy3)（x>0，y>0）解答：化簡(jiǎn)為x√(xy)x√(xy)+2y√(xy)=2y√(xy)5.化簡(jiǎn)：(√(3)+1)2√(12)+(√(3)1)?1解答：展開(kāi)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、絕對(duì)值化簡(jiǎn)1.化簡(jiǎn)：|x5|+|x+3|（分情況討論）解答：關(guān)鍵點(diǎn)x=5和x=-3，①x≥5時(shí)，原式=(x-5)+(x+3)=2x-2；②-3≤x<5時(shí)，原式=(5-x)+(x+3)=8；③x<-3時(shí)，原式=(5-x)+(-x-3)=2-2x2.化簡(jiǎn)：|2a1||a+2|（a為實(shí)數(shù)，分情況討論）解答：關(guān)鍵點(diǎn)a=1/2和a=-2，①a≥1/2時(shí)，原式=(2a-1)-(a+2)=a-3；②-2≤a<1/2時(shí)，原式=(1-2a)-(a+2)=-3a-1；③a<-2時(shí)，原式=(1-2a)-(-a-2)=-a+33.化簡(jiǎn)：|m4|2|m+1|（當(dāng)m≥4時(shí)；當(dāng)-1≤m<4時(shí)；當(dāng)m<-1時(shí)）解答：①m≥4時(shí)，原式=(m-4)-2(m+1)=-m-6；②-1≤m<4時(shí)，原式=(4-m)-2(m+1)=4-m-2m-2=2-3m；③m<-1時(shí)，原式=(4-m)-2(-m-1)=4-m+2m+2=m+64.化簡(jiǎn)：|3t|+|t5|（t為實(shí)數(shù)，求最小值及對(duì)應(yīng)t值）解答：當(dāng)3≤t≤5時(shí)，原式=(t-3)+(5-t)=2，為最小值；t<3時(shí)，原式=(3-t)+(5-t)=8-2t>2；t>5時(shí)，原式=(t-3)+(t-5)=2t-8>2，故最小值為2，t∈[3,5]5.化簡(jiǎn)：|x24||x2|（x>1，分情況討論）解答：x2-4=(x-2)(x+2)，關(guān)鍵點(diǎn)x=2（x>1），①1<x<2時(shí)，原式=(4-x2)-(2-x)=4-x2-2+x=-x2+x+2；②x≥2時(shí)，原式=(x2-4)-(x-2)=x2-x-2五、指數(shù)運(yùn)算化簡(jiǎn)1.化簡(jiǎn)：(2a2b?3)3×(a?1b2)?2÷(4a?2b?1)解答：計(jì)算得8a?b??×a2b??÷4a?2b?1=2a??2?2b?????1=2a1?b?12=2a1?/b122.化簡(jiǎn)：(3x?2y3)2÷(2x3y??)?1×(6x?1y?2)解答：計(jì)算得9x??y?÷(x?3y?/2)×6x?1y?2=9x??y?×2x3y?×6x?1y?2=108x???3?1y????2=108x?2y?=108y?/x23.化簡(jiǎn)：[(2?1a3b?2)2×(4a?2b3)?1]÷(8a??b?)?2解答：計(jì)算得(2?2a?b??×4?1a2b?3)÷(8?2a?b?1?)=(2?2×2?2a??2b???3)÷(2??a?b?1?)=(2??a?b??)×2?a??b1?=22b3=4b34.化