Further problems 2

Exercise 1

Simplify:

  1. −27ab3a\frac{-27ab}{3a}

  2. 3(a+b)(a−b)6(a+b)\frac{3(a+b)(a-b)}{6(a+b)}

  3. 14ab+7ac+42ab70ab+14ac+7ab\frac{14ab+7ac+42ab}{70ab+14ac+7ab}

  4. −(3a+n)(b−c)c−b-\frac{(3a+n)(b-c)}{c-b}

  5. ax+ay3x+3y\frac{ax+ay}{3x+3y}

  6. ax+xbx+x\frac{ax+x}{bx+x}

  7. 2x−410(6x−12)\frac{2x-4}{10(6x-12)}

  8. −4xz2bxz−2axz\frac{-4xz}{2bxz-2axz}

  9. −15b+15a−25(b−a)\frac{-15b+15a}{-25(b-a)}

  10. 8x−8yx2−y2\frac{8x-8y}{x^2-y^2}

  11. 6am−9an−4bm+6bn3a−2b\frac{6am-9an-4bm+6bn}{3a-2b}

  12. 6a2−6ab3a2−6ab+3b2\frac{6a^2-6ab}{3a^2-6ab+3b^2}

  13. 2x2−18y22x2+12xy+18y2\frac{2x^2-18y^2}{2x^2+12xy+18y^2}

Solution
  • −27ab3a=−9b‾\frac{-27ab}{3a}=\underline{-9b}
  • 3(a+b)(a−b)6(a+b)=a−b2‾\frac{3(a+b)(a-b)}{6(a+b)}=\underline{\frac{a-b}{2}}
  • 14ab+7ac+42ab70ab+14ac+7ab=56ab+7ac77ab+14ac=7a(8b+c)7a(11b+2c)=8b+c11b+2c‾\frac{14ab+7ac+42ab}{70ab+14ac+7ab}= \frac{56ab+7ac}{77ab+14ac}= \frac{7a(8b+c)}{7a(11b+2c)}=\underline{\frac{8b+c}{11b+2c}}
  • −(3a+n)(b−c)c−b=(3a+n)(b−c)−(c−b)=(3a+n)(b−c)b−c=3a+n‾-\frac{(3a+n)(b-c)}{c-b}=\frac{(3a+n)(b-c)}{-(c-b)}= \frac{(3a+n)(b-c)}{b-c}=\underline{3a+n}
  • ax+ay3x+3y=a(x+y)3(x+y)=a3‾\frac{ax+ay}{3x+3y}=\frac{a(x+y)}{3(x+y)}=\underline{\frac{a}{3}}
  • ax+xbx+x=(a+1)x(b+1)x=a+1b+1‾\frac{ax+x}{bx+x}=\frac{(a+1)x}{(b+1)x}=\underline{\frac{a+1}{b+1}}
  • 2x−410(6x−12)=2(x−2)60(x−2)=130‾\frac{2x-4}{10(6x-12)}=\frac{2(x-2)}{60(x-2)}=\underline{\frac{1}{30}}
  • −4xz2bxz−2axz=−4xz2xz(b−a)=−2b−a=2−(b−a)=2a−b‾\frac{-4xz}{2bxz-2axz}=\frac{-4xz}{2xz(b-a)}=\frac{-2}{b-a}=\frac{2}{-(b-a)}=\underline{\frac{2}{a-b}}
  • −15b+15a−25(b−a)=15(a−b)25(a−b)=35‾\frac{-15b+15a}{-25(b-a)}=\frac{15(a-b)}{25(a-b)}=\underline{\frac{3}{5}}
  • 8x−8yx2−y2=8(x−y)(x+y)(x−y)=8x+y‾\frac{8x-8y}{x^2-y^2}=\frac{8(x-y)}{(x+y)(x-y)}=\underline{\frac{8}{x+y}}
  • 6am−9an−4bm+6bn3a−2b=6am−4bm−9an+6bn3a−2b=2m(3a−2b)−3n(3a−2b)3a−2b=2m−3n‾\frac{6am-9an-4bm+6bn}{3a-2b} = \frac{6am-4bm \quad -9an + 6bn}{3a-2b}= \frac{2m(3a-2b) - 3n(3a-2b)}{3a-2b} = \underline{2m-3n}
  • 6a2−6ab3a2−6ab+3b2=6a(a−b)3(a2−2ab+b2)=2a(a−b)(a−b)2=2aa−b‾\frac{6a^2-6ab}{3a^2-6ab+3b^2}=\frac{6a(a-b)}{3(a^2-2ab+b^2)} = \frac{2a(a-b)}{(a-b)^2}=\underline{\frac{2a}{a-b}}
  • 2x2−18y22x2+12xy+18y2=2(x2−9y2)2(x2+6xy+9y2)=(x+3y)(x−3y)(x+3y)2=x−3yx+3y‾\frac{2x^2-18y^2}{2x^2+12xy+18y^2}=\frac{2(x^2-9y^2)}{2(x^2+6xy+9y^2)} = \frac{(x+3y)(x-3y)}{(x+3y)^2}=\underline{\frac{x-3y}{x+3y}}
    • Exercise 2

    Simplify:

    1. 3⋅8133 \cdot \frac{8}{13}

    2. −244⋅11\frac{-2}{44} \cdot 11

    3. 2x⋅38xy2x \cdot \frac{3}{8xy}

    4. b+1a+1⋅(1+a)\frac{b+1}{a+1} \cdot (1+a)

    5. 6ab−4cd12ab+8cd⋅(6ab+4cd)\frac{6ab-4cd}{12ab+8cd} \cdot (6ab+4cd)

    6. 127⋅146\frac{12}{7} \cdot \frac{14}{6}

    7. 310⋅−710\frac{3}{10} \cdot \frac{-7}{10}

    8. −513⋅39−18-\frac{5}{13} \cdot \frac{39}{-18}

    9. 3xy7ad⋅14cd8yz\frac{3xy}{7ad} \cdot \frac{14cd}{8yz}

    10. 1−a1−n⋅a+11−a\frac{1-a}{1-n} \cdot \frac{a+1}{1-a}

    11. 5(y−a)6ax⋅4(a+y)y−a⋅3bx\frac{5(y-a)}{6ax} \cdot \frac{4(a+y)}{y-a} \cdot 3bx

    12. 3ab⋅(−bc)⋅xc\frac{3a}{b} \cdot (-bc) \cdot \frac{x}{c}

    13. 2x−y10x⋅6yx−2y⋅5x3y\frac{2x-y}{10x} \cdot \frac{6y}{x-2y} \cdot \frac{5x}{3y}

    14. 4a+812b−6⋅3a−64b+2⋅5+10ba+2\frac{4a+8}{12b-6} \cdot \frac{3a-6}{4b+2} \cdot \frac{5+10b}{a+2}

  • 3⋅813=3⋅813=2413‾3 \cdot \frac{8}{13}=\frac{3 \cdot 8}{13}=\underline{\frac{24}{13}}
  • −244⋅11=−2⋅1144=−12‾\frac{-2}{44} \cdot 11=-\frac{2 \cdot 11}{44}=\underline{-\frac{1}{2}}
  • 2x⋅38xy=2x⋅38xy=34y‾2x \cdot \frac{3}{8xy}=\frac{2x \cdot 3}{8xy}=\underline{\frac{3}{4y}}
  • b+1a+1⋅(1+a)=(b+1)(a+1)a+1=b+1‾\frac{b+1}{a+1} \cdot (1+a)= \frac{(b+1)(a+1)}{a+1}=\underline{b+1}
  • 6ab−4cd12ab+8cd⋅(6ab+4cd)=2(3ab−2cd)⋅2(3ab+2cd)4(3ab+2cd)=3ab−2cd‾\frac{6ab-4cd}{12ab+8cd} \cdot (6ab+4cd) = \frac{2(3ab-2cd)\cdot 2(3ab+2cd)}{4(3ab+2cd)} = \underline{3ab-2cd}
  • 127⋅146=12⋅147⋅6=4‾\frac{12}{7} \cdot \frac{14}{6}=\frac{12 \cdot 14}{7 \cdot 6}=\underline{4}
  • 310⋅−710=−21100‾\frac{3}{10} \cdot \frac{-7}{10}=\underline{-\frac{21}{100}}
  • −513⋅39−18=5⋅3913⋅18=5⋅318=56‾-\frac{5}{13} \cdot \frac{39}{-18}=\frac{5 \cdot 39}{13 \cdot 18}=\frac{5 \cdot 3}{18}=\underline{\frac{5}{6}}
  • 3xy7ad⋅14cd8yz=3xy7ad⋅7cd4yz=3⋅7xycd7⋅4adyz=3xc4az‾\frac{3xy}{7ad} \cdot \frac{14cd}{8yz}=\frac{3xy}{7ad} \cdot \frac{7cd}{4yz} =\frac{3 \cdot 7 x y c d}{7 \cdot 4 a d y z}=\underline{\frac{3xc}{4az}}
  • 1−a1−n⋅a+11−a=a+11−n‾\frac{1-a}{1-n} \cdot \frac{a+1}{1-a}=\underline{\frac{a+1}{1-n}}
  • 5(y−a)6ax⋅4(a+y)y−a⋅3bx=60(y−a)(a+y)bx6ax(y−a)=10(a+y)ba‾\frac{5(y-a)}{6ax} \cdot \frac{4(a+y)}{y-a} \cdot 3bx =\frac{60(y-a)(a+y)bx}{6ax(y-a)}=\underline{\frac{10(a+y)b}{a}}
  • 3ab⋅(−bc)⋅xc=−3abcxbc=−3ax‾\frac{3a}{b} \cdot (-bc) \cdot \frac{x}{c}=-\frac{3a b c x}{bc}=\underline{-3ax}
  • 2x−y10x⋅6yx−2y⋅5x3y=(2x−y)30xy30xy(x−2y)=2x−yx−2y‾\frac{2x-y}{10x} \cdot \frac{6y}{x-2y} \cdot \frac{5x}{3y}=\frac{(2x-y)30xy}{30xy(x-2y)} =\underline{\frac{2x-y}{x-2y}}
  • 4a+812b−6⋅3a−64b+2⋅5+10ba+2=4(a+2)⋅3(a−2)⋅5(1+2b)6(2b−1)⋅2(2b+1)⋅(a+2)=5(a−2)2b−1‾\frac{4a+8}{12b-6} \cdot \frac{3a-6}{4b+2} \cdot \frac{5+10b}{a+2} = \frac{4(a+2) \cdot 3(a-2) \cdot 5(1+2b)}{6(2b-1) \cdot 2(2b+1) \cdot (a+2)} = \underline{\frac{5(a-2)}{2b-1}}
    • Q3
    Exercise 3

    Simplify

    1. 98:5\frac{9}{8} : 5

    2. 5:785 : \frac{7}{8}

    3. 1219:738\frac{12}{19} : \frac{7}{38}

    4. −29:(−1927)-\frac{2}{9} : (-\frac{19}{27})

    5. 35x:7\frac{35}{x} : 7

    6. 12a7x:(4ax)\frac{12a}{7x} : (4ax)

    7. 18p:9p5x18p : \frac{9p}{5x}

    8. 12ab25x:24ay5cx\frac{12ab}{25x} : \frac{24ay}{5cx}

    9. a+55b:a−510c\frac{a+5}{5b} : \frac{a-5}{10c}

    10. a+1 a2−1a  \frac{a+1}{\,\frac{a^2-1}{a}\;}

    Solution

    todo

    Q4
    Exercise 4

    Write as a fraction and simplify the result:

    1. 23+56\frac{2}{3} + \frac{5}{6}

    2. 34−13\frac{3}{4} - \frac{1}{3}

    3. 74+85−110\frac{7}{4} + \frac{8}{5} - \frac{1}{10}

    4. 5b4+3b4\frac{5b}{4} + \frac{3b}{4}

    5. 3x2b−x2b\frac{3x}{2b} - \frac{x}{2b}

    6. 6aa−b−15ba−b+9aa−b\frac{6a}{a-b} - \frac{15b}{a-b} +\frac{9a}{a-b}

    7. 10a[3b−2(3a−2b)]a(m−n)−5a[2b+3(2a−3b)]a(m−n)\frac{10a[3b-2(3a-2b)]}{a(m-n)} \\ - \frac{5a[2b+3(2a-3b)]}{a(m-n)}

    8. 3a−4b4+a+6b3−7a+b6\frac{3a-4b}{4}+\frac{a+6b}{3}-\frac{7a+b}{6}

    9. 2m15ab−3n5a\frac{2m}{15ab}-\frac{3n}{5a}

    10. 5y−4x18y−3x+2y4xy\frac{5y-4x}{18y}-\frac{3x+2y}{4xy}

    11. 2v9ab−3u2ac+5w6abc\frac{2v}{9ab} - \frac{3u}{2ac} + \frac{5w}{6abc}

    12. 1−2x−3y2x1-\frac{2x-3y}{2x}

    13. 5x+730ab+2−9x15bc+6x−1512ac\frac{5x+7}{30ab} + \frac{2-9x}{15bc} + \frac{6x-15}{12ac}

    14. 4x+4y6a−2−x+3y3a−1+4y9a−3\frac{4x+4y}{6a-2} - \frac{x+3y}{3a-1} + \frac{4y}{9a-3}

    15. 2x+y8x−4y+2x−y6x+3y−3xy4x2−y2\frac{2x+y}{8x-4y}+\frac{2x-y}{6x+3y}-\frac{3xy}{4x^2-y^2}

    16. (3x−2)24−9x2+27x2−1227x2−36x+12\frac{(3x-2)^2}{4-9x^2}+\frac{27x^2-12}{27x^2-36x+12}

    17. 7a−2a21−4a2+a+12a−1−2a2a+1\frac{7a-2a^2}{1-4a^2} + \frac{a+1}{2a-1} - \frac{2a}{2a+1}

    Solution

    todo

    Q5
    Exercise 5

    Simplify

    1. 12x21+x−2x\frac{1}{\frac{2x^2}{1+x}-2x}

    2. 1x+1y1x−1y\frac{\frac{1}{x}+\frac{1}{y}}{\frac{1}{x}-\frac{1}{y}}

    3. 2−4x2yy−x2y\frac{2-\frac{4x}{2y}}{y-\frac{x^2}{y}}

    4. 4a2−4ab+1b21a−12b\frac{\frac{4}{a^2}-\frac{4}{ab}+\frac{1}{b^2}}{\frac{1}{a}-\frac{1}{2b}}

    5. aa+a1−aa−x\frac{a}{a+\frac{a}{1-\frac{a}{a-x}}}

    6. (1a−1a+1)2\left(\frac{1}{a}-\frac{1}{a+1}\right)^2

    7. aa2−11a+1−1a−1\frac{\frac{a}{a^2-1}}{\frac{1}{a+1}-\frac{1}{a-1}}

    8. a−b15x−a−b12ya−b5x−a−b4y\frac{\frac{a-b}{15x}-\frac{a-b}{12y}}{\frac{a-b}{5x}-\frac{a-b}{4y}}

    9. 1−6x+9x29x2−1\frac{1-\frac{6}{x}+\frac{9}{x^2}}{\frac{9}{x^2}-1}

    Solution

    todo

    Show

    Solutions

    A1
    A2
    A3
    1. 98:5=9851=98⋅15=940‾\frac{9}{8} : 5=\frac{\frac{9}{8}}{\frac{5}{1}} = \frac{9}{8}\cdot \frac{1}{5}=\underline{\frac{9}{40}}
    2. 5:78=5178=51⋅87=407‾5 : \frac{7}{8}=\frac{\frac{5}{1}}{\frac{7}{8}} = \frac{5}{1} \cdot \frac{8}{7}=\underline{\frac{40}{7}}
    3. 1219:738=1219738=1219⋅387=12⋅3819⋅7=12⋅27=247‾\frac{12}{19} : \frac{7}{38} = \frac{\frac{12}{19}}{\frac{7}{38}} = \frac{12}{19} \cdot \frac{38}{7}=\frac{12 \cdot 38}{19 \cdot 7} =\frac{12 \cdot 2}{7}=\underline{\frac{24}{7}}
    4. −29:(−1927)=−29−1927=29⋅2719=2⋅319=619‾-\frac{2}{9} : (-\frac{19}{27})=\frac{-\frac{2}{9}}{-\frac{19}{27}} = \frac{2}{9} \cdot \frac{27}{19}=\frac{2 \cdot 3}{19}=\underline{\frac{6}{19}}
    5. 35x:7=35x71=35x⋅17=5x‾\frac{35}{x} : 7 = \frac{\frac{35}{x}}{\frac{7}{1}} = \frac{35}{x}\cdot \frac{1}{7}= \underline{\frac{5}{x}}
    6. 12a7x:(4ax)=12a7x4ax1=12a7x⋅14ax=12a7x⋅4ax=37x2‾\frac{12a}{7x} : (4ax) = \frac{\frac{12a}{7x}}{\frac{4ax}{1}} = \frac{12a}{7x} \cdot \frac{1}{4ax}=\frac{12a}{7x \cdot 4ax}=\underline{\frac{3}{7x^2}}
    7. 18p:9p5x=18p19p5x=18p1⋅5x9p=18p⋅5x9p=10x‾18p : \frac{9p}{5x} = \frac{\frac{18p}{1}}{\frac{9p}{5x}} = \frac{18p}{1} \cdot \frac{5x}{9p}=\frac{18p \cdot 5x}{9p}=\underline{10x}
    8. 12ab25x:24ay5cx=12ab25x24ay5cx=12ab25x⋅5cx24ay=b⋅c5⋅2y=bc10y‾\frac{12ab}{25x} : \frac{24ay}{5cx} = \frac{\frac{12ab}{25x}}{\frac{24ay}{5cx}} = \frac{12ab}{25x} \cdot \frac{5cx}{24ay} = \frac{b \cdot c}{5\cdot 2y} =\underline{\frac{bc}{10y}}
    9. a+55b:a−510c=a+55ba−510c=a+55b⋅10ca−5=2c(a+5)b(a−5)‾\frac{a+5}{5b} : \frac{a-5}{10c}=\frac{\frac{a+5}{5b}}{\frac{a-5}{10c}} = \frac{a+5}{5b} \cdot \frac{10c}{a-5}=\underline{\frac{2c(a+5)}{b(a-5)}}
    10. a+1 a2−1a  =a+11a2−1a=a+11⋅a(a+1)(a−1)=aa−1‾\frac{a+1}{\,\frac{a^2-1}{a}\;}=\frac{\frac{a+1}{1}}{\frac{a^2-1}{a}} = \frac{a+1}{1} \cdot \frac{a}{(a+1)(a-1)}=\underline{\frac{a}{a-1}}
    A4

    1. 23+56=46+56=96=32‾\frac{2}{3} + \frac{5}{6} = \frac{4}{6}+\frac{5}{6}=\frac{9}{6}=\underline{\frac{3}{2}}

    2. 34−13=912−412=512‾\frac{3}{4} - \frac{1}{3}=\frac{9}{12}-\frac{4}{12}=\underline{\frac{5}{12}}

    3. 74+85−110=3520+3220−220=6520=134‾\frac{7}{4} + \frac{8}{5} - \frac{1}{10}=\frac{35}{20}+\frac{32}{20}-\frac{2}{20}=\frac{65}{20}=\underline{\frac{13}{4}}

    4. 5b4+3b4=8b4=2b‾\frac{5b}{4} + \frac{3b}{4}=\frac{8b}{4}=\underline{2b}

    5. 3x2b−x2b=2x2b=xb‾\frac{3x}{2b} - \frac{x}{2b}=\frac{2x}{2b}=\underline{\frac{x}{b}}

    6. 6aa−b−15ba−b+9aa−b=6a−15b+9aa−b=15a−15ba−b=15(a−b)a−b=15‾\frac{6a}{a-b} - \frac{15b}{a-b} +\frac{9a}{a-b}=\frac{6a-15b+9a}{a-b}=\frac{15a-15b}{a-b} = \frac{15(a-b)}{a-b}=\underline{15}

    7. 10a[3b−2(3a−2b)]a(m−n)−5a[2b+3(2a−3b)]a(m−n)=10[3b−2(3a−2b)]m−n−5[2b+3(2a−3b)]m−n=10[3b−2(3a−2b)]−5[2b+3(2a−3b)]m−n=5⋅2[3b−2(3a−2b)]−[2b+3(2a−3b)]m−n=5⋅6b−4(3a−2b)−2b−3(2a−3b)m−n=5⋅6b−12a+8b−2b−6a+9bm−n=5⋅21b−18am−n=5⋅3(7b−6a)m−n=15(7b−6a)m−n‾\frac{10a[3b-2(3a-2b)]}{a(m-n)} - \frac{5a[2b+3(2a-3b)]}{a(m-n)}\\=\frac{10[3b-2(3a-2b)]}{m-n}-\frac{5[2b+3(2a-3b)]}{m-n}\\= \frac{10[3b-2(3a-2b)]-5[2b+3(2a-3b)]}{m-n}\\=5\cdot\frac{2[3b-2(3a-2b)]-[2b+3(2a-3b)]}{m-n} \\= 5\cdot\frac{6b-4(3a-2b)-2b-3(2a-3b)}{m-n}\\=5\cdot\frac{6b-12a+8b-2b-6a+9b}{m-n} \\= 5\cdot\frac{21b-18a}{m-n}\\=5\cdot\frac{3(7b-6a)}{m-n}=\underline{\frac{15(7b-6a)}{m-n}}

    8. 3a−4b4+a+6b3−7a+b6=3(3a−4b)12+4(a+6b)12−2(7a+b)12=9a−12b+4a+24b−14a−2b12=10b−a12‾\frac{3a-4b}{4}+\frac{a+6b}{3}-\frac{7a+b}{6}\\=\frac{3(3a-4b)}{12}+\frac{4(a+6b)}{12}-\frac{2(7a+b)}{12}\\=\frac{9a-12b+4a+24b-14a-2b}{12}\\=\underline{\frac{10b-a}{12}}

    9. 2m15ab−3n5a=2m15ab−9bn15ab=2m−9bn15ab‾\frac{2m}{15ab}-\frac{3n}{5a}=\frac{2m}{15ab}-\frac{9bn}{15ab}=\underline{\frac{2m-9bn}{15ab}}

    10. 5y−4x18y−3x+2y4xy=2x(5y−4x)36xy−9(3x+2y)36xy=10xy−8x2−27x−18y36xy‾\frac{5y-4x}{18y}-\frac{3x+2y}{4xy}=\frac{2x(5y-4x)}{36xy}-\frac{9(3x+2y)}{36xy}=\underline{\frac{10xy-8x^2-27x-18y}{36xy}}

    11. 2v9ab−3u2ac+5w6abc=2c⋅2v18abc−9b⋅3u18abc+3⋅5w18abc=4cv−27bu+15w18abc‾\frac{2v}{9ab} - \frac{3u}{2ac} + \frac{5w}{6abc}= \frac{2c \cdot 2v}{18abc} - \frac{9b \cdot 3u}{18abc} + \frac{3 \cdot 5w}{18abc}= \underline{\frac{4cv - 27bu + 15w}{18abc}}

    12. 1−2x−3y2x=2x2x−2x−3y2x=2x−(2x−3y)2x=2x−2x+3y2x=3y2x‾1-\frac{2x-3y}{2x}=\frac{2x}{2x}-\frac{2x-3y}{2x}=\frac{2x-(2x-3y)}{2x}=\frac{2x-2x+3y}{2x}=\underline{\frac{3y}{2x}}or: …=1−(2x2x−3y2x)=1−1+3y2x=3y2x‾\quad \ldots = 1-\left(\frac{2x}{2x}-\frac{3y}{2x}\right)=1-1+\frac{3y}{2x}=\underline{\frac{3y}{2x}}

    13. 5x+730ab+2−9x15bc+6x−1512ac=2c(5x+7)60abc+4a(2−9x)60abc+5b(6x−15)60abc=10cx+14c+8a−36ax+30bx−75b60abc‾\frac{5x+7}{30ab} + \frac{2-9x}{15bc} + \frac{6x-15}{12ac}= \frac{2c(5x+7)}{60abc} + \frac{4a(2-9x)}{60abc} + \frac{5b(6x-15)}{60abc}\\= \underline{\frac{10cx+14c+8a-36ax+30bx-75b}{60abc}}

    14. 4x+4y6a−2−x+3y3a−1+4y9a−3=4x+4y2(3a−1)−x+3y3a−1+4y3(3a−1)=3(4x+4y)6(3a−1)−6(x+3y)6(3a−1)+2⋅4y6(3a−1)=12x+12y−6x−18y+8y6(3a−1)=6x+2y6(3a−1)=3x+y3(3a−1)‾\frac{4x+4y}{6a-2} - \frac{x+3y}{3a-1} + \frac{4y}{9a-3}= \frac{4x+4y}{2(3a-1)} - \frac{x+3y}{3a-1} + \frac{4y}{3(3a-1)}\\=\frac{3(4x+4y)}{6(3a-1)} - \frac{6(x+3y)}{6(3a-1)} + \frac{2 \cdot 4y}{6(3a-1)}\\ = \frac{12x+12y-6x-18y+8y}{6(3a-1)}\\=\frac{6x+2y}{6(3a-1)}\\= \underline{\frac{3x+y}{3(3a-1)}}

    15. 2x+y8x−4y+2x−y6x+3y−3xy4x2−y2=2x+y4(2x−y)+2x−y3(x+2y)−3xy(2x+y)(2x−y)=3(2x+y)212(2x+y)(2x−y)+4(2x−y)212(2x+y)(2x−y)−12⋅3xy12(2x+y)(2x−y)=3(4x2+4xy+y2)+4(4x2−4xy+y2)−36xy12(2x+y)(2x−y)=12x2+12xy+3y2+16x2−16xy+4y2−36xy12(2x+y)(2x−y)=28x2−40xy+7y212(4x2−y2)‾\frac{2x+y}{8x-4y}+\frac{2x-y}{6x+3y}-\frac{3xy}{4x^2-y^2}\\=\frac{2x+y}{4(2x-y)} + \frac{2x-y}{3(x+2y)}-\frac{3xy}{(2x+y)(2x-y)} \\=\frac{3(2x+y)^2}{12(2x+y)(2x-y)} + \frac{4(2x-y)^2}{12(2x+y)(2x-y)} - \frac{12 \cdot 3xy}{12(2x+y)(2x-y)} \\= \frac{3(4x^2+4xy+y^2)+4(4x^2-4xy+y^2)-36xy}{12(2x+y)(2x-y)} \\= \frac{12x^2+12xy+3y^2+16x^2-16xy+4y^2-36xy}{12(2x+y)(2x-y)}=\underline{\frac{28x^2-40xy+7y^2}{12(4x^2-y^2)}}

    16. (3x−2)24−9x2+27x2−1227x2−36x+12=(3x−2)24−9x2+3(9x2−4)3(9x2−12x+4)=(3x−2)2(2+3x)(2−3x)+3(3x+2)(3x−2)3(3x−2)2=(3x−2)2−(3x+2)(3x−2)+3x+23x−2=−(3x−2)2(3x+2)(3x−2)+(3x+2)2(3x+2)(3x−2)=−(9x2−12x+4)+(9x2+12x+4)(3x+2)(3x−2)=24x9x2−4‾\frac{(3x-2)^2}{4-9x^2}+\frac{27x^2-12}{27x^2-36x+12}=\frac{(3x-2)^2}{4-9x^2} + \frac{3(9x^2-4)}{3(9x^2-12x+4)}\\=\frac{(3x-2)^2}{(2+3x)(2-3x)} + \frac{3(3x+2)(3x-2)}{3(3x-2)^2}\\ = \frac{(3x-2)^2}{-(3x+2)(3x-2)} + \frac{3x+2}{3x-2} \\= -\frac{(3x-2)^2}{(3x+2)(3x-2)} + \frac{(3x+2)^2}{(3x+2)(3x-2)} \\=\frac{-(9x^2-12x+4)+(9x^2+12x+4)}{(3x+2)(3x-2)} = \underline{\frac{24x}{9x^2-4}}

    17. 7a−2a21−4a2+a+12a−1−2a2a+1=7a−2a2(1−2a)(1+2a)+a+12a−1−2a2a+1=7a−2a2−(2a−1)(2a+1)+(a+1)(2a+1)(2a−1)(2a+1)−(2a−1)2a(2a−1)(2a+1)=−(7a−2a2)+2a2+a+2a+1−(4a2−2a)(2a−1)(2a+1)=−7a+2a2+2a2+3a+1−4a2+2a(2a−1)(2a+1)=1−2a−(1−2a)(2a+1)=−12a+1‾\frac{7a-2a^2}{1-4a^2} + \frac{a+1}{2a-1} - \frac{2a}{2a+1}=\frac{7a-2a^2}{(1-2a)(1+2a)} + \frac{a+1}{2a-1} - \frac{2a}{2a+1}\\= \frac{7a-2a^2}{-(2a-1)(2a+1)} + \frac{(a+1)(2a+1)}{(2a-1)(2a+1)} - \frac{(2a-1)2a}{(2a-1)(2a+1)} \\= \frac{-(7a-2a^2)+2a^2+a+2a+1-(4a^2-2a)}{(2a-1)(2a+1)} \\= \frac{-7a+2a^2+2a^2+3a+1-4a^2+2a}{(2a-1)(2a+1)} \\= \frac{1-2a}{-(1-2a)(2a+1)}\\=\underline{-\frac{1}{2a+1}}

    A5
    1. 12x21+x−2x=12x21+x−2x(1+x)1+x=12x2−2x−2x21+x=−1+x2x‾\frac{1}{\frac{2x^2}{1+x}-2x}=\frac{1}{\frac{2x^2}{1+x}-\frac{2x(1+x)}{1+x}}= \frac{1}{\frac{2x^2-2x-2x^2}{1+x}}=\underline{-\frac{1+x}{2x}}
    2. 1x+1y1x−1y=(1x+1y)xy(1x−1y)xy=y+xy−x‾\frac{\frac{1}{x}+\frac{1}{y}}{\frac{1}{x}-\frac{1}{y}}=\frac{\left(\frac{1}{x}+\frac{1}{y}\right)xy}{\left(\frac{1}{x}-\frac{1}{y}\right)xy}= \underline{\frac{y+x}{y-x}}
    3. 2−4x2yy−x2y=2(1−xy)y−x2y=2(1−xy)y(y−x2y)y=2(y−x)y2−x2=2(y−x)(y+x)(y−x)=2y+x‾\frac{2-\frac{4x}{2y}}{y-\frac{x^2}{y}}=\displaystyle \frac{2\left(1-\frac{x}{y}\right)}{y-\frac{x^2}{y}}=\frac{2\left(1-\frac{x}{y}\right)y}{\left(y-\frac{x^2}{y}\right)y} = \frac{2(y-x)}{y^2-x^2} = \frac{2(y-x)}{(y+x)(y-x)}=\underline{\frac{2}{y+x}}
    4. 4a2−4ab+1b21a−12b=4b2a2b2−4aba2b2+a2a2b22b2ab−a2ab=  4b2−4ab+a2a2b2  2b−a2ab=(2b−a)2a2b2⋅2ab2b−a=2(2b−a)ab‾\frac{\frac{4}{a^2}-\frac{4}{ab}+\frac{1}{b^2}}{\frac{1}{a}-\frac{1}{2b}} = \frac{\frac{4b^2}{a^2b^2}-\frac{4ab}{a^2b^2}+\frac{a^2}{a^2b^2}}{\frac{2b}{2ab}-\frac{a}{2ab}}= \frac{\;\frac{4b^2-4ab+a^2}{a^2b^2}\;}{\frac{2b-a}{2ab}}= \frac{(2b-a)^2}{a^2b^2} \cdot \frac{2ab}{2b-a} = \underline{\frac{2(2b-a)}{ab}}
    5. aa+a1−aa−x=aa+aa−xa−x−aa−x=aa+a −xa−x  =aa+a⋅a−x−x=aa+ax−ax=aaxx+ax−a2x=aax+ax−a2x=ax2ax−a2=axa(2x−a)=x2x−a‾\frac{a}{a+\frac{a}{1-\frac{a}{a-x}}} = \frac{a}{a+\frac{a}{\frac{a-x}{a-x}-\frac{a}{a-x}}} = \frac{a}{a+\frac{a}{\,\frac{-x}{a-x}\;}}=\frac{a}{a+a\cdot\frac{a-x}{-x}}\\= \frac{a}{a+a\frac{x-a}{x}}\\ = \frac{a}{\frac{ax}{x}+\frac{ax-a^2}{x}}\\ = \frac{a}{\frac{ax+ax-a^2}{x}}\\ = a\frac{x}{2ax-a^2}\\ = \frac{ax}{a(2x-a)}\\=\underline{\frac{x}{2x-a}}
    6. (1a−1a+1)2=(a+1a(a+1)−aa(a+1))2=(a+1−aa(a+1))2=1a2(a+1)2=1(a2+a)2‾\left(\frac{1}{a}-\frac{1}{a+1}\right)^2=\left(\frac{a+1}{a(a+1)}-\frac{a}{a(a+1)}\right)^2 = \left(\frac{a+1-a}{a(a+1)}\right)^2=\underline{\frac{1}{a^2(a+1)^2}=\frac{1}{(a^2+a)^2}}
    7. aa2−11a+1−1a−1=(a(a+1)(a−1))(a+1)(a−1)(1a+1−1a−1)(a+1)(a−1)=a(a−1)−(a+1)=a−2=−a2‾\frac{\frac{a}{a^2-1}}{\frac{1}{a+1}-\frac{1}{a-1}}=\frac{\left(\frac{a}{(a+1)(a-1)}\right)(a+1)(a-1)}{\left(\frac{1}{a+1}-\frac{1}{a-1}\right)(a+1)(a-1)}=\frac{a}{(a-1)-(a+1)}=\frac{a}{-2}=\underline{-\frac{a}{2}}
    8. a−b15x−a−b12ya−b5x−a−b4y=4y(a−b)60xy−5x(a−b)60xy4y(a−b)20xy−5x(a−b)20xy=(4y−5x)(a−b)60xy(4y−5x)(a−b)20xy=(4y−5x)(a−b)60xy⋅20xy(4y−5x)(a−b)=13‾\frac{\frac{a-b}{15x}-\frac{a-b}{12y}}{\frac{a-b}{5x}-\frac{a-b}{4y}}=\frac{\frac{4y(a-b)}{60xy}-\frac{5x(a-b)}{60xy}}{\frac{4y(a-b)}{20xy}-\frac{5x(a-b)}{20xy}}=\frac{\frac{(4y-5x)(a-b)}{60xy}}{\frac{(4y-5x)(a-b)}{20xy}}=\frac{(4y-5x)(a-b)}{60xy} \cdot \frac{20xy}{(4y-5x)(a-b)}=\underline{\frac{1}{3}}
    9. 1−6x+9x29x2−1=(1−6x+9x2)x2(9x2−1)x2=x2−6x+99−x2=(x−3)2(3+x)(3−x)=(3−x)2(3+x)(3−x)=3−x3+x‾\frac{1-\frac{6}{x}+\frac{9}{x^2}}{\frac{9}{x^2}-1}=\frac{\left(1-\frac{6}{x}+\frac{9}{x^2}\right)x^2}{\left(\frac{9}{x^2}-1\right)x^2}=\frac{x^2-6x+9}{9-x^2}=\frac{(x-3)^2}{(3+x)(3-x)}=\frac{(3-x)^2}{(3+x)(3-x)}=\underline{\frac{3-x}{3+x}}