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Aufgabe

  1. Vereinfache soweit wie möglich
    1. 12d9d\frac{12d}{9d}
    2. 16xyz−20xz\frac{16xyz}{-20xz}
    3. 24a2bc256ab3c\frac{24a^2bc^2}{56ab^3c}
    4. 6a+206a\frac{6a+20}{6a}
    5. p3−p2p3+p2\frac{p^3-p^2}{p^3+p^2}
    6. rs−rtsu−tu\frac{rs-rt}{su-tu}
    7. a2−aab+a\frac{a^2-a}{ab+a}
    8. a−bb−a\frac{a-b}{b-a}
    9. 3a+3ba2+b2\frac{3a+3b}{a^2+b^2}
    10. 4mp−2030−6mp\frac{4mp-20}{30-6mp}
    11. a2+2ab+b2a+b\frac{a^2+2ab+b^2}{a+b}
    12. u2+2uv+v24u+4v\frac{u^2+2uv+v^2}{4u+4v}
    13. u2−v2u−v\frac{u^2-v^2}{u-v}
    14. e2−e1−e2\frac{e^2-e}{1-e^2}
    15. (4s2+25)(2s+5)16s4−625\frac{(4s^2+25)(2s+5)}{16s^4-625}
    16. a2+2a−24a2−6a+8\frac{a^2+2a-24}{a^2-6a+8}
  2. Schreibe als einen Bruch und vereinfache soweit wie möglich
    1. 4u21+9u14\frac{4u}{21}+\frac{9u}{14}
    2. 4u21⋅9u14\frac{4u}{21}\cdot\frac{9u}{14}
    3. 4u21:9u14\frac{4u}{21}:\frac{9u}{14}
    4. zn2+43n\frac{z}{n^2}+\frac{4}{3n}
    5. a+bb−a−ba\frac{a+b}{b}-\frac{a-b}{a}
    6. x+y2xy+y+z2yz+x+z2xz\frac{x+y}{2xy}+\frac{y+z}{2yz}+\frac{x+z}{2xz}
    7. a3+1\frac{a}{3}+1
    8. 5w−1+3w5w-1+\frac{3}{w}
    9. 3−mm−n3-\frac{m}{m-n}
    10. x−y15x+10y+x+y3x+2y\frac{x-y}{15x+10y}+\frac{x+y}{3x+2y}
    11. 8ss2−4+2+ss−2\frac{8s}{s^2-4}+\frac{2+s}{s-2}
  3. Schreibe als einen Bruch und vereinfache soweit wie möglich
    1. 8a3b⋅9bc4a\frac{8a}{3b}\cdot\frac{9bc}{4a}
    2. (3x+3y)⋅9cx+y(3x+3y)\cdot\frac{9c}{x+y}
    3. 5q2−1⋅(q−1)\frac{5}{q^2-1}\cdot(q-1)
    4. d−118d⋅12d21−d\frac{d-1}{18d}\cdot\frac{12d^2}{1-d}
    5. x2−6xy+9y25m−5n⋅m2−n2x−3y\frac{x^2-6xy+9y^2}{5m-5n}\cdot\frac{m^2-n^2}{x-3y}
    6. g+13g−13\frac{g+\frac{1}{3}}{g-\frac{1}{3}}
    7. 1−nv−nv\frac{1-\frac{n}{v}}{-nv}
    8. ab−cd1d−1b\frac{\frac{a}{b}-\frac{c}{d}}{\frac{1}{d}-\frac{1}{b}}
    9. ef⋅ghef−gh\frac{\frac{e}{f}\cdot\frac{g}{h}}{\frac{e}{f}-\frac{g}{h}}
    10. 1−1e1−1e2\frac{1-\frac{1}{e}}{1-\frac{1}{e^2}}
    11. x+2x2−2x−84x+8\frac{\frac{x+2}{x^2-2x-8}}{4x+8}
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Lösung

  1. Vereinfache soweit wie möglich
    1. 12d9d=43\frac{12d}{9d}=\frac{4}{3}
    2. 16xyz−20xz=−4y5\frac{16xyz}{-20xz}=-\frac{4y}{5}
    3. 24a2bc256ab3c=3ac14b2\frac{24a^2bc^2}{56ab^3c}=\frac{3ac}{14 b^2}
    4. 6a+206a=3a+103a\frac{6a+20}{6a}=\frac{3a+10}{3a}
    5. p3−p2p3+p2=p−1p+1\frac{p^3-p^2}{p^3+p^2}=\frac{p-1}{p+1}
    6. rs−rtsu−tu=r(s−t)u(s−t)=ru\frac{rs-rt}{su-tu}=\frac{r(s-t)}{u(s-t)}=\frac{r}{u}
    7. a2−aab+a=a(a−1)a(b+1)=a−1b+1\frac{a^2-a}{ab+a}=\frac{a(a-1)}{a(b+1)}=\frac{a-1}{b+1}
    8. a−bb−a=1⋅(a−b)−1⋅(a−b)=1−1=−1\frac{a-b}{b-a}=\frac{1\cdot (a-b)}{-1\cdot(a-b)}=\frac{1}{-1}=-1
    9. 3a+3ba2+b2\frac{3a+3b}{a^2+b^2} lässt sich nicht vereinfachen
    10. 4mp−2030−6mp=4(mp−5)−6(mp−5)=−23\frac{4mp-20}{30-6mp}=\frac{4(mp-5)}{-6(mp-5)}=-\frac{2}{3}
    11. a2+2ab+b2a+b=(a+b)2(a+b)=a+b\frac{a^2+2ab+b^2}{a+b}=\frac{(a+b)^2}{(a+b)}=a+b
    12. u2+2uv+v24u+4v=(u+v)24(u+v)=u+v4\frac{u^2+2uv+v^2}{4u+4v}=\frac{(u+v)^2}{4(u+v)}=\frac{u+v}{4}
    13. u2−v2u−v=(u+v)(u−v)(u−v)=u+v\frac{u^2-v^2}{u-v}=\frac{(u+v)(u-v)}{(u-v)}=u+v
    14. e2−e1−e2=e(e−1)(1−e)(1+e)=−e(1−e)(1−e)(1+e)=−e1+e\frac{e^2-e}{1-e^2}=\frac{e(e-1)}{(1-e)(1+e)}=\frac{-e(1-e)}{(1-e)(1+e)}=-\frac{e}{1+e}
    15. (4s2+25)(2s+5)16s4−625=(4s2+25)(2s+5)(4s2−25)(4s2+25)=2s+5(2s−5)(2s+5)=12s−5\frac{(4s^2+25)(2s+5)}{16s^4-625}=\frac{(4s^2+25)(2s+5)}{(4s^2-25)(4s^2+25)}=\frac{2s+5}{(2s-5)(2s+5)}=\frac{1}{2s-5}
    16. a2+2a−24a2−6a+8=(a+6)(a−4)(a−4)(a−2))=a+6a−2\frac{a^2+2a-24}{a^2-6a+8}=\frac{(a+6)(a-4)}{(a-4)(a-2))}=\frac{a+6}{a-2}
  2. Schreibe als einen Bruch und vereinfache soweit wie möglich
    1. 4u21+9u14=8u42+27u42=35u42=5u6\frac{4u}{21}+\frac{9u}{14}=\frac{8u}{42}+\frac{27u}{42}=\frac{35u}{42}=\frac{5u}{6}
    2. 4u21⋅9u14=4u⋅9u21⋅14=2⋅3⋅2u⋅3u3⋅7⋅2⋅7=6u249\frac{4u}{21}\cdot\frac{9u}{14}=\frac{4u\cdot 9u}{21\cdot 14}=\frac{2\cdot 3\cdot 2u\cdot 3u}{3\cdot 7\cdot 2\cdot 7}=\frac{6u^2}{49}
    3. 4u21:9u14=4u21⋅149u=2⋅2u⋅2⋅73⋅7⋅3⋅3u=827\frac{4u}{21}:\frac{9u}{14}=\frac{4u}{21}\cdot\frac{14}{9u}=\frac{2\cdot 2u\cdot 2\cdot 7}{3\cdot 7\cdot 3\cdot 3u}=\frac{8}{27}
    4. z+nn2+43n=3(n+z)+4n3n2=7n+3z3n2\frac{z+n}{n^2}+\frac{4}{3n}=\frac{3(n+z)+4n}{3n^2}=\frac{7n+3z}{3n^2}
    5. a+bb−a−ba=a(a+b)−b(a−b)ab=a2+ab−ab+b2ab=a2+b2ab\frac{a+b}{b}-\frac{a-b}{a}=\frac{a(a+b)-b(a-b)}{ab}=\frac{a^2+ab-ab+b^2}{ab}=\frac{a^2+b^2}{ab}
    6. x+y2xy+y+z2yz+x+z2xz=z(x+y)+x(y+z)+y(x+z)2xyz=2(xy+xz+yz)2xyz=xy+xz+yzxyz\frac{x+y}{2xy}+\frac{y+z}{2yz}+\frac{x+z}{2xz}=\frac{z(x+y)+x(y+z)+y(x+z)}{2xyz}=\frac{2(xy+xz+yz)}{2xyz}=\frac{xy+xz+yz}{xyz}
    7. a3+1=a3+33=a+33\frac{a}{3}+1=\frac{a}{3}+\frac{3}{3}=\frac{a+3}{3}
    8. 5w−1+3w=w(5w−1)+3w=5w2−w+3w5w-1+\frac{3}{w}=\frac{w(5w-1)+3}{w}=\frac{5w^2-w+3}{w}
    9. 3−mm−n=3(m−n)m−n−mm−n=2m−3nm−n3-\frac{m}{m-n}=\frac{3(m-n)}{m-n}-\frac{m}{m-n}=\frac{2m-3n}{m-n}
    10. x−y15x+10y+x+y3x+2y=x−y15x+10y+5(x+y)15x+10y=6x+4y15x+10y\frac{x-y}{15x+10y}+\frac{x+y}{3x+2y}=\frac{x-y}{15x+10y}+\frac{5(x+y)}{15x+10y}=\frac{6x+4y}{15x+10y}
    11. 8ss2−4+2+ss−2=8s(s−2)(s+2)+(2+s)(s+2)(s−2)(s+2)=s2+12s+4(s−2)(s+2)\frac{8s}{s^2-4}+\frac{2+s}{s-2}=\frac{8s}{(s-2)(s+2)}+\frac{(2+s)(s+2)}{(s-2)(s+2)}=\frac{s^2+12s+4}{(s-2)(s+2)}
  3. Vereinfache soweit wie möglich
    1. 8a3b⋅9bc4a=6c\frac{8a}{3b}\cdot\frac{9bc}{4a}=6c
    2. (3x+3y)⋅9cx+y=3(x+y)1⋅9cx+y=27c(3x+3y)\cdot\frac{9c}{x+y}=\frac{3(x+y)}{1}\cdot \frac{9c}{x+y}=27c
    3. 5q2−1⋅(q−1)=5(q−1)(q+1)⋅q−11=5q+1\frac{5}{q^2-1}\cdot(q-1)=\frac{5}{(q-1)(q+1)}\cdot \frac{q-1}{1}=\frac{5}{q+1}
    4. d−118d⋅12d21−d=d−118d⋅12d2−1⋅(d−1)=−12d218d=−2d3\frac{d-1}{18d}\cdot\frac{12d^2}{1-d}=\frac{d-1}{18d}\cdot\frac{12d^2}{-1\cdot(d-1)}=-\frac{12d^2}{18d}=-\frac{2d}{3}
    5. x2−6xy+9y25m−5n⋅m2−n2x−3y=(x−3y)2(m−n)(m+n)5(m−n)(x−3y)=(x−3y)(m+n)5\frac{x^2-6xy+9y^2}{5m-5n}\cdot\frac{m^2-n^2}{x-3y}=\frac{(x-3y)^2 (m-n)(m+n)}{5(m-n)(x-3y)}=\frac{(x-3y)(m+n)}{5}
    6. g+13g−13=3g+133g−13=3g+13⋅33g−1=3g+13g−1\frac{g+\frac{1}{3}}{g-\frac{1}{3}}=\frac{\frac{3g+1}{3}}{\frac{3g-1}{3}}=\frac{3g+1}{3}\cdot\frac{3}{3g-1}=\frac{3g+1}{3g-1}
    7. 1−nv−nv=v−nv−nv1=−v−nv⋅1nv=−v−nnv2=n−vnv2\frac{1-\frac{n}{v}}{-nv}=\frac{\frac{v-n}{v}}{\frac{-nv}{1}}=-\frac{v-n}{v}\cdot\frac{1}{nv}=-\frac{v-n}{n v^2}=\frac{n-v}{n v^2}
    8. ab−cd1d−1b=ad−bcbdb−dbd=ad−bcb−d\frac{\frac{a}{b}-\frac{c}{d}}{\frac{1}{d}-\frac{1}{b}}=\frac{\frac{ad-bc}{bd}}{\frac{b-d}{bd}}=\frac{ad-bc}{b-d}
    9. ef⋅ghef−gh=egfheh−fgfh=egeh−fg\frac{\frac{e}{f}\cdot\frac{g}{h}}{\frac{e}{f}-\frac{g}{h}}=\frac{\frac{eg}{fh}}{\frac{eh-fg}{fh}}=\frac{eg}{eh-fg}
    10. 1−1e1−1e2=e−1ee2−1e2=e−1e⋅e2(e−1)(e+1)=ee+1\frac{1-\frac{1}{e}}{1-\frac{1}{e^2}}=\frac{\frac{e-1}{e}}{\frac{e^2-1}{e^2}}=\frac{e-1}{e}\cdot\frac{e^2}{(e-1)(e+1)}=\frac{e}{e+1}
    11. x+2x2−2x−84x+8=x+2(x−4)(x+2)4x+8=1x−44x+81=1x−4⋅14x+8=14(x+2)(x−4)\frac{\frac{x+2}{x^2-2x-8}}{4x+8}=\frac{\frac{x+2}{(x-4)(x+2)}}{4x+8}=\frac{\frac{1}{x-4}}{\frac{4x+8}{1}}=\frac{1}{x-4}\cdot \frac{1}{4x+8}=\frac{1}{4(x+2)(x-4)}