7.3E: Mazoezi

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Mazoezi hufanya kamili

Ongeza na Ondoa Maneno ya busara na Denominator ya kawaida

Katika mazoezi yafuatayo, ongeza.

1. $\dfrac{2}{15}+\dfrac{7}{15}$

Jibu: $\dfrac{3}{5}$

2. $\dfrac{7}{24}+\dfrac{11}{24}$

3. $\dfrac{3c}{4c−5}+\dfrac{5}{4c−5}$

Jibu: $\dfrac{3c+5}{4c−5}$

4. $\dfrac{7m}{2m+n}+\dfrac{4}{2m+n}$

5. $\dfrac{2r^2}{2r−1}+\dfrac{15r−8}{2r−1}$

Jibu: $r+8$

6. $\dfrac{3s^2}{3s−2}+\dfrac{13s−10}{3s−2}$

7. $\dfrac{2w^2}{w^2−16}+\dfrac{8w}{w^2−16}$

Jibu: $\dfrac{2w}{w−4}$

8. $\dfrac{7x^2}{x^2−9}+\dfrac{21x}{x^2−9}$

Katika mazoezi yafuatayo, toa.

9. $\dfrac{9a^2}{3a−7}−\dfrac{49}{3a−7}$

Jibu: $3a+7$

10. $\dfrac{25b^2}{5b−6}−\dfrac{36}{5b−6}$

11. $\dfrac{3m^2}{6m−30}−\dfrac{21m−30}{6m−30}$

Jibu: $\dfrac{m−2}{2}$

12. $\dfrac{2n^2}{4n−32}−\dfrac{18n−16}{4n−32}$

13. $\dfrac{6p^2+3p+4}{p^2+4p−5}−\dfrac{5p^2+p+7}{p^2+4p−5}$

Jibu: $\dfrac{p+3}{p+5}$

14. $\dfrac{5q^2+3q−9}{q^2+6q+8}−\dfrac{4q^2+9q+7}{q^2+6q+8}$

15. $\dfrac{5r^2+7r−33}{r^2−49}−\dfrac{4r^2+5r+30}{r^2−49}$

Jibu: $\dfrac{r+9}{r+7}$

16. $\dfrac{7t^2−t−4}{t^2−25}−\dfrac{6t^2+12t−44}{t^2−25}$

Kuongeza na Ondoa Maneno ya busara ambao Denominators ni kinyume

Katika mazoezi yafuatayo, ongeza au uondoe.

17. $\dfrac{10v}{2v−1}+\dfrac{2v+4}{1−2v}$

Jibu: $4$

18. $\dfrac{20w}{5w−2}+\dfrac{5w+6}{2−5w}$

19. $\dfrac{10x^2+16x−7}{8x−3}+\dfrac{2x^2+3x−1}{3−8x}$

Jibu: $x+2$

20. $\dfrac{6y^2+2y−11}{3y−7}+\dfrac{3y^2−3y+17}{7−3y}$

21. $\dfrac{z^2+6z}{z^2−25}−\dfrac{3z+20}{25−z^2}$

Jibu: $\dfrac{z+4}{z−5}$

22. $\dfrac{a^2+3a}{a^2−9}−\dfrac{3a−27}{9−a^2}$

23. $\dfrac{2b^2+30b−13}{b^2−49}−\dfrac{2b^2−5b−8}{49−b^2}$

Jibu: $\dfrac{4b−3}{b−7}$

24. $\dfrac{c^2+5c−10}{c^2−16}−\dfrac{c^2−8c−10}{16−c^2}$

Pata Denominator ya kawaida ya maneno ya busara

Katika mazoezi yafuatayo, a. kupata LCD kwa maneno yaliyotolewa ya busara b. kuandika tena kama maneno sawa ya busara na denominator ya chini ya kawaida.

25. $\dfrac{5}{x^2−2x−8},\dfrac{2x}{x^2−x−12}$

Jibu: a.$(x+2)(x−4)(x+3)$
b.$\dfrac{5x+15}{(x+2)(x−4)(x+3)}$,
$\dfrac{2x^2+4x}{(x+2)(x−4)(x+3)}$

26. $\dfrac{8}{y^2+12y+35},\dfrac{3y}{y^2+y−42}$

27. $\dfrac{9}{z^2+2z−8},\dfrac{4z}{z^2−4}$

Jibu: a.$(z−2)(z+4)(z−4)$
b.$\dfrac{9z−36}{(z−2)(z+4)(z−4)}$,
$\dfrac{4z^2−8z}{(z−2)(z+4)(z−4)}$

28. $\dfrac{6}{a^2+14a+45},\dfrac{5a}{a^2−81}$

29. $\dfrac{4}{b^2+6b+9},\dfrac{2b}{b^2−2b−15}$

Jibu: a.$(b+3)(b+3)(b−5)$
b.$\dfrac{4b−20}{(b+3)(b+3)(b−5)}$,
$\dfrac{2b^2+6b}{(b+3)(b+3)(b−5)}$

30. $\dfrac{5}{c^2−4c+4},\dfrac{3c}{c^2−7c+10}$

31. $\dfrac{2}{3d^2+14d−5},\dfrac{5d}{3d^2−19d+6}$

Jibu: a.$(d+5)(3d−1)(d−6)$
b.$\dfrac{2d−12}{(d+5)(3d−1)(d−6)}$,
$\dfrac{5d^2+25d}{(d+5)(3d−1)(d−6)}$

32. $\dfrac{3}{5m^2−3m−2},\dfrac{6m}{5m^2+17m+6}$

Kuongeza na Ondoa Maneno ya busara na Tofauti na Denominators

Katika mazoezi yafuatayo, fanya shughuli zilizoonyeshwa.

33. $\dfrac{7}{10x^2y}+\dfrac{4}{15xy^2}$

Jibu: $\dfrac{21y+8x}{30x^2y^2}$

34. $\dfrac{1}{12a^3b^2}+\dfrac{5}{9a^2b^3}$

35. $\dfrac{3}{r+4}+\dfrac{2}{r−5}$

Jibu: $\dfrac{5r−7}{(r+4)(r−5)}$

36. $\dfrac{4}{s−7}+\dfrac{5}{s+3}$

37. $\dfrac{5}{3w−2}+\dfrac{2}{w+1}$

Jibu: $\dfrac{11w+1}{(3w−2)(w+1)}$

38. $\dfrac{4}{2x+5}+\dfrac{2}{x−1}$

39. $\dfrac{2y}{y+3}+\dfrac{3}{y−1}$

Jibu: $\dfrac{2y^2+y+9}{(y+3)(y−1)}$

40. $\dfrac{3z}{z−2}+\dfrac{1}{z+5}$

41. $\dfrac{5b}{a^2b−2a^2}+\dfrac{2b}{b^2−4}$

Jibu: $\dfrac{b(5b+10+2a^2)}{a^2(b−2)(b+2)}$

42. $\dfrac{4}{cd+3c}+\dfrac{1}{d^2−9}$

43. $\dfrac{−3m}{3m−3}+\dfrac{5m}{m^2+3m−4}$

Jibu: $-\dfrac{m}{m+4}$

44. $\dfrac{8}{4n+4}+\dfrac{6}{n^2−n−2}$

45. $\dfrac{3r}{r^2+7r+6}+\dfrac{9}{r^2+4r+3}$

Jibu: $\dfrac{3(r^2+6r+18)}{(r+1)(r+6)(r+3)}$

46. $\dfrac{2s}{s^2+2s−8}+\dfrac{4}{s^2+3s−10}$

47. $\dfrac{t}{t−6}−\dfrac{t−2}{t+6}$

Jibu: $\dfrac{2(7t−6)}{(t−6)(t+6)}$

48. $\dfrac{x−3}{x+6}−\dfrac{x}{x+3}$

49. $\dfrac{5a}{a+3}−\dfrac{a+2}{a+6}$

Jibu: $\dfrac{4a^2+25a−6}{(a+3)(a+6)}$

50. $\dfrac{3b}{b−2}−\dfrac{b−6}{b−8}$

51. $\dfrac{6}{m+6}−\dfrac{12m}{m^2−36}$

Jibu: $\dfrac{−6}{m−6}$

52. $\dfrac{4}{n+4}−\dfrac{8n}{n^2−16}$

53. $\dfrac{−9p−17}{p^2−4p−21}−\dfrac{p+1}{7−p}$

Jibu: $\dfrac{p+2}{p+3}$

54. $\dfrac{−13q−8}{q^2+2q−24}−\dfrac{q+2}{4−q}$

55. $\dfrac{−2r−16}{r^2+6r−16}−\dfrac{5}{2−r}$

Jibu: $\dfrac{3}{r−2}$

56. $\dfrac{2t−30}{t^2+6t−27}−\dfrac{2}{3−t}$

57. $\dfrac{2x+7}{10x−1}+3$

Jibu: $\dfrac{4(8x+1)}{10x−1}$

58. $\dfrac{8y−4}{5y+2}−6$

59. $\dfrac{3}{x^2−3x−4}−\dfrac{2}{x^2−5x+4}$

Jibu: $\dfrac{x−5}{(x−4)(x+1)(x−1)}$

60. $\dfrac{4}{x^2−6x+5}−\dfrac{3}{x^2−7x+10}$

61. $\dfrac{5}{x^2+8x−9}−\dfrac{4}{x^2+10x+9}$

Jibu: $\dfrac{1}{(x−1)(x+1)}$

62. $\dfrac{3}{2x^2+5x+2}−\dfrac{1}{2x^2+3x+1}$

63. $\dfrac{5a}{a−2}+\dfrac{9}{a}−\dfrac{2a+18}{a^2−2a}$

Jibu: $\dfrac{5a^2+7a−36}{a(a−2)}$

64. $\dfrac{2b}{b−5}+\dfrac{3}{2b}−\dfrac{2b−15}{2b^2−10b}$

65. $\dfrac{c}{c+2}+\dfrac{5}{c−2}−\dfrac{10c}{c^2−4}$

Jibu: $\dfrac{c−5}{c+2}$

66. $\dfrac{6d}{d−5}+\dfrac{1}{d+4}+\dfrac{7d−5}{d^2−d−20}$

67. $\dfrac{3d}{d+2}+\dfrac{4}{d}−\dfrac{d+8}{d^2+2d}$

Jibu: $\dfrac{3(d+1)}{d+2}$

68. $\dfrac{2q}{q+5}+\dfrac{3}{q−3}−\dfrac{13q+15}{q^2+2q−15}$

Ongeza na Ondoa Kazi za busara

Katika mazoezi yafuatayo, tafuta.$R(x)=f(x)+g(x)$ b$R(x)=f(x)−g(x)$.

69. $f(x)=\dfrac{−5x−5}{x^2+x−6}$na$ g(x)=\dfrac{x+1}{2−x}$

Jibu: a.$R(x)=−\dfrac{(x+8)(x+1)}{(x−2)(x+3)}$
b.$R(x)=\dfrac{x+1}{x+3}$

70. $f(x)=\dfrac{−4x−24}{x^2+x−30}$na$ g(x)=\dfrac{x+7}{5−x}$

71. $f(x)=\dfrac{6x}{x^2−64}$na$g(x)=\dfrac{3}{x−8}$

Jibu: a.$R(x)=\dfrac{3(3x+8)}{(x−8)(x+8)}$
b.$R(x)=\dfrac{3}{x+8}$

72. $f(x)=\dfrac{5}{x+7}$na$ g(x)=\dfrac{10x}{x^2−49}$

Mazoezi ya kuandika

73. Donald anadhani kuwa$\dfrac{3}{x}+\dfrac{4}{x}$ ni$\dfrac{7}{2x}$. Donald ni sahihi? Eleza.

Jibu: Majibu yatatofautiana.

74. Eleza jinsi unavyopata Denominator ya kawaida ya$x^2+5x+4$ na$x^2−16$.

75. Felipe anadhani$\dfrac{1}{x}+\dfrac{1}{y}$ ni$\dfrac{2}{x+y}$.
Chagua maadili ya namba kwa x na y na tathmini$\dfrac{1}{x}+\dfrac{1}{y}$.
b Tathmini$\dfrac{2}{x+y}$ kwa maadili sawa ya x na y uliyotumia katika sehemu a..
Eleza kwa nini Felipe ni makosa.
d. kupata kujieleza sahihi kwa ajili ya$1x+1y$.

Jibu: a Majibu yatatofautiana.
b Majibu yatatofautiana.
c Majibu yatatofautiana.
d.$\dfrac{x+y}{x}$

76. Kurahisisha maneno$\dfrac{4}{n^2+6n+9}−\dfrac{1}{n^2−9}$ na kuelezea hatua zako zote.

Self Check

Baada ya kukamilisha mazoezi, tumia orodha hii ili kutathmini ujuzi wako wa malengo ya sehemu hii.

$Jedwali hili lina nguzo nne na safu sita. Mstari wa kwanza ni kichwa na huandika kila safu, “Siwezi...”, “Kwa ujasiri,” “Kwa msaada fulani,” na “Hakuna-Siipati!” Katika mstari wa 2, naweza mara kuongeza na kuondoa maneno ya busara na denominator ya kawaida. mfululizo 3, naweza mara kuongeza na Ondoa maneno ya busara na denominators kwamba ni kinyume. mfululizo 4, naweza kupata denominator angalau ya kawaida ya maneno ya busara. mfululizo 5, naweza mara kuongeza na Ondoa maneno ya busara na denominators tofauti. mfululizo 6, naweza mara kuongeza au Ondoa kazi busara. Hakuna kitu katika nguzo nyingine.$

b Baada ya kuchunguza orodha hii, utafanya nini ili uwe na ujasiri kwa malengo yote?