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7.2E: Mazoezi

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    Mazoezi hufanya kamili

    Tambua Maadili ambayo Ufafanuzi wa busara haukufafanuliwa

    Katika mazoezi yafuatayo, tambua maadili ambayo maneno ya busara hayajafafanuliwa.

    1. a.\(\dfrac{2x^2}{z}\) b.\(\dfrac{4p−1}{6p−5}\) c.\(\dfrac{n−3}{n^2+2n−8}\)

    Jibu

    a.\(z=0\)
    b.\(p=\dfrac{5}{6}\)
    c.\(n=−4, n=2\)

    2. a.\(\dfrac{10m}{11n}\) b.\(\dfrac{6y+13}{4y−9}\) c.\(\dfrac{b−8}{b^2−36}\)

    3. a.\(\dfrac{4x^2y}{3y}\) b.\(\dfrac{3x−2}{2x+1}\) c.\(\dfrac{u−1}{u^2−3u−28}\)

    Jibu

    a.\(y=0\)
    b.\(x=−\dfrac{1}{2}\)
    c.\(u=−4, u=7\)

    4. a.\(\dfrac{5pq^2}{9q}\) b.\(\dfrac{7a−4}{3a+5}\) c.\(\dfrac{1}{x^2−4}\)

    Kurahisisha maneno ya busara

    Katika mazoezi yafuatayo, kurahisisha kila kujieleza kwa busara.

    5. \(−\dfrac{44}{55}\)

    Jibu

    \(−\dfrac{4}{5}\)

    6. \(\dfrac{56}{63}\)

    7. \(\dfrac{8m^3n}{12mn^2}\)

    Jibu

    \(\dfrac{2m^2}{3n}\)

    8. \(\dfrac{36v^3w^2}{27vw^3}\)

    9. \(\dfrac{8n−96}{3n−36}\)

    Jibu

    \(\dfrac{8}{3}\)

    10. \(\dfrac{12p−240}{5p−100}\)

    11. \(\dfrac{x^2+4x−5}{x^2−2x+1}\)

    Jibu

    \(\dfrac{x+5}{x−1}\)

    12. \(\dfrac{y^2+3y−4}{y^2−6y+5}\)

    13. \(\dfrac{a^2−4}{a^2+6a−16}\)

    Jibu

    \(\dfrac{a+2}{a+8}\)

    14. \(\dfrac{y^2−2y−3}{y^2−9}\)

    15. \(\dfrac{p^3+3p^2+4p+12}{p^2+p−6}\)

    Jibu

    \(\dfrac{p^2+4}{p−2}\)

    16. \(\dfrac{x^3−2x^2−25x+50}{x^2−25}\)

    17. \(\dfrac{8b^2−32b}{2b^2−6b−80}\)

    Jibu

    \(\dfrac{4b(b−4)}{(b+5)(b−8)}\)

    18. \(\dfrac{−5c^2−10c}{−10c^2+30c+100}\)

    19. \(\dfrac{3m^2+30mn+75n^2}{4m^2−100n^2}\)

    Jibu

    \(\dfrac{3(m+5n)}{4(m−5n)}\)

    20. \(\dfrac{5r^2+30rs−35s^2}{r^2−49s^2}\)

    21. \(\dfrac{a−5}{5−a}\)

    Jibu

    \(−1\)

    22. \(\dfrac{5−d}{d−5}\)

    23. \(\dfrac{20−5y}{y^2−16}\)

    Jibu

    \(\dfrac{−5}{y+4}\)

    24. \(\dfrac{4v−32}{64−v^2}\)

    25. \(\dfrac{w^3+21}{6w^2−36}\)

    Jibu

    \(\dfrac{w^2−6w+3}{6w−6}\)

    26. \(\dfrac{v^3+125}{v^2−25}\)

    27. \(\dfrac{z^2−9z+20}{16−z^2}\)

    Jibu

    \(\dfrac{−z−5}{4+z}\)

    28. \(\dfrac{a^2−5z−36}{81−a^2}\)

    Kuzidisha maneno ya busara

    Katika mazoezi yafuatayo, kuzidisha maneno ya busara.

    29. \(\dfrac{12}{16}·\dfrac{4}{10}\)

    Jibu

    \(\dfrac{3}{10}\)

    30. \(\dfrac{3}{25}·\dfrac{16}{24}\)

    31. \(\dfrac{5x^2y^4}{12xy^3}·\dfrac{6x^2}{20y^2}\)

    Jibu

    \(\dfrac{x^3}{8y}\)

    32. \(\dfrac{12a^3b}{b^2}·\dfrac{2ab^2}{9b^3}\)

    33. \(\dfrac{5p^2}{p^2−5p−36}·\dfrac{p^2−16}{10p}\)

    Jibu

    \(\dfrac{p(p−4)}{2(p−9)}\)

    34. \(\dfrac{3q^2}{q^2+q−6}·\dfrac{q^2−9}{9q}\)

    35. \(\dfrac{2y^2−10y}{y^2+10y+25}·\dfrac{y+5}{6y}\)

    Jibu

    \(\dfrac{y−5}{3(y+5)}\)

    36. \(\dfrac{z^2+3z}{z^2−3z−4}·\dfrac{z−4}{z^2}\)

    37. \(\dfrac{28−4b}{3b−3}·\dfrac{b^2+8b−9}{b^2−49}\)

    Jibu

    \(\dfrac{−4(b+9)}{3(b+7)}\)

    38. \(\dfrac{72m−12m^2}{8m+32}·\dfrac{m^2+10m+24}{m^2−36}\)

    39. \(\dfrac{c^2-10c+25}{c^2−25}·\dfrac{c^2+10c+25}{3c^2−14c−5}\)

    Jibu

    \(\dfrac{c+5}{3c+1}\)

    40. \(\dfrac{2d^2+d−3}{d^2−16}·\dfrac{d^2−8d+16}{2d^2−9d−18}\)

    41. \(\dfrac{2m^2−3m−2}{2m2+7m+3}·\dfrac{3m^2−14m+15}{3m^2+17m−20}\)

    Jibu

    \(\dfrac{(m−3)(m−2)}{(m+4)(m+3)}\)

    42. \(\dfrac{2n^2−3n−14}{25−n^2}·\dfrac{n^2−10n+25}{2n^2−13n+21}\)

    Gawanya Maneno ya busara

    Katika mazoezi yafuatayo, fungua maneno ya busara.

    43. \(\dfrac{v−5}{11−v}÷\dfrac{v^2−25}{v−11}\)

    Jibu

    \(−\dfrac{1}{v+5}\)

    44. \(\dfrac{10+w}{w−8}÷\dfrac{100−w^2}{8−w}\)

    45. \(\dfrac{3s^2}{s^2−16}÷\dfrac{s^3−4s^2+16s}{s^3−64}\)

    Jibu

    \(\dfrac{3s}{s+4}\)

    46. \(\dfrac{r^2−9}{15}÷\dfrac{r^3−27}{5r^2+15r+45}\)

    47. \(\dfrac{p^3+q^3}{3p^2+3pq+3q^2}÷\dfrac{p^2−q^2}{12}\)

    Jibu

    \(\dfrac{4(p^2−pq+q^2)}{(p−q)(p^2+pq+q^2)}\)

    48. \(\dfrac{v^3−8w^3}{2v^2+4vw+8w^2}÷\dfrac{v^2−4w^2}{4}\)

    49. \(\dfrac{x^2+3x−10}{4x}÷(2x^2+20x+50)\)

    Jibu

    \(\dfrac{x−2}{8x(x+5)}\)

    50. \(\dfrac{2y^2−10yz−48z^2}{2y−1}÷(4y^2−32yz)\)

    51. \(\dfrac{\dfrac{2a^2−a−21}{5a+20}}{\dfrac{a^2+7a+12}{a^2+8a+16}}\)

    Jibu

    \(\dfrac{2a−7}{5}\)

    52. \(\dfrac{\dfrac{3b^2+2b−8}{12b+18}}{\dfrac{3b^2+2b−8}{2b^2−7b−15}}\)

    53. \(\dfrac{\dfrac{12c^2−12}{2c^2−3c+1}}{\dfrac{4c+4}{6c^2−13c+5}}\)

    Jibu

    \(3(3c−5)\)

    54. \(\dfrac{\dfrac{4d^2+7d−2}{35d+10}}{\dfrac{d^2−4}{7d^2−12d−4}}\)

    Kwa mazoezi yafuatayo, fanya shughuli zilizoonyeshwa.

    55. \(\dfrac{10m^2+80m}{3m−9}·\dfrac{m^2+4m−21}{m^2−9m+20}÷\dfrac{5m^2+10m}{2m−10}\)

    Jibu

    \(\dfrac{4(m+8)(m+7)}{3(m−4)(m+2)}\)

    56. \(\dfrac{4n^2+32n}{3n+2}·\dfrac{3n^2−n−2}{n^2+n−30}÷\dfrac{108n^2−24n}{n+6}\)

    57. \(\dfrac{12p^2+3p}{p+3}÷\dfrac{p^2+2p−63}{p^2−p−12}·\dfrac{p−7}{9p^3−9p^2}\)

    Jibu

    \(\dfrac{(4p+1)(p−4)}{3p(p+9)(p−1)}\)

    58. \(\dfrac{6q+3}{9q^2−9q}÷\dfrac{q^2+14q+33}{q^2+4q−5}·\dfrac{4q^2+12q}{12q+6}\)

    Kuzidisha na Kugawanya Kazi

    Katika mazoezi yafuatayo, tafuta uwanja wa kila kazi.

    59. \(R(x)=\dfrac{x^3−2x^2−25x+50}{x^2−25}\)

    Jibu

    \(x\neq 5\)na\(x\neq −5\)

    60. \(R(x)=\dfrac{x^3+3x^2−4x−12}{x^2−4}\)

    61. \(R(x)=\dfrac{3x^2+15x}{6x^2+6x−36}\)

    Jibu

    \(x\neq 2\)na\(x\neq −3\)

    62. \(R(x)=\dfrac{8x^2−32x}{2x^2−6x−80}\)

    Kwa mazoezi yafuatayo, tafuta\(R(x)=f(x)·g(x)\) wapi\(f(x)\) na\(g(x)\) unapewa.

    63. \(f(x)=\dfrac{6x^2−12x}{x^2+7x−18} \quad g(x)=\dfrac{x^2−81}{3x^2−27x}\)

    Jibu

    \(R(x)=2\)

    64. \(f(x)=\dfrac{x^2−2x}{x^2+6x−16} \quad g(x)=\dfrac{x^2−64}{x^2−8x}\)

    65. \(f(x)=\dfrac{4x}{x^2−3x−10} \quad g(x)=\dfrac{x^2−25}{8x^2}\)

    Jibu

    \(R(x)=\dfrac{x+5}{2x(x+2)}\)

    66. \(f(x)=\dfrac{2x^2+8x}{x^2−9x+20} \quad g(x)=\dfrac{x−5}{x^2}\)

    Kwa mazoezi yafuatayo, tafuta\(R(x)=f(x)g(x)\) wapi\(f(x)\) na\(g(x)\) unapewa.

    67. \(f(x)=\dfrac{27x^2}{3x−21} \quad g(x)=\dfrac{3x^2+18x}{x^2+13x+42}\)

    Jibu

    \(R(x)=\dfrac{3x(x+7)}{x−7}\)

    68. \(f(x)=\dfrac{24x^2}{2x−8} \quad g(x)=\dfrac{4x^3+28x^2}{x^2+11x+28}\)

    69. \(f(x)=\dfrac{16x^2}{4x+36} \quad g(x)=\dfrac{4x^2−24x}{x^2+4x−45}\)

    Jibu

    \(R(x)=\dfrac{x(x−5)}{x−6}\)

    70. \(f(x)=\dfrac{24x^2}{2x−4} \quad g(x)=\dfrac{12x^2+36x}{x^2−11x+18}\)

    Mazoezi ya kuandika

    71. Eleza jinsi unavyopata maadili ya x ambayo kujieleza kwa busara\(\dfrac{x^2−x−20}{x^2−4}\) haijulikani.

    Jibu

    Majibu yatatofautiana.

    72. Eleza hatua zote unazochukua ili kurahisisha kujieleza kwa busara\(\dfrac{p^2+4p−21}{9−p^2}\).

    73. a. kuzidisha\(\dfrac{7}{4}·\dfrac{9}{10}\) na kueleza hatua yako yote.
    b Kuzidisha\(\dfrac{n}{n−3}·\dfrac{9}{n+3}\) na kuelezea hatua zako zote.
    c Tathmini jibu lako kwa sehemu b. wakati\(n=7\). Je, kupata jibu moja wewe got katika sehemu a.? Kwa nini au kwa nini?

    Jibu

    Majibu yatatofautiana.

    74. a. Gawanya\(\dfrac{24}{5}÷6\) na kuelezea hatua zako zote.
    B. kugawanya\(\dfrac{x^2−1}{x}÷(x+1)\) na kueleza hatua yako yote.
    c Tathmini jibu lako kwa sehemu b. wakati\(x=5\). Je, kupata jibu moja wewe got katika sehemu a.? Kwa nini au kwa nini?

    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 kuamua maadili ambayo kujieleza kwa busara haijulikani. mfululizo 3, naweza mara kurahisisha maneno mantiki. Katika mstari wa 4, naweza mara kuzidisha maneno ya busara. Katika mstari wa 5, naweza mara kugawanya maneno ya busara. Katika mstari wa 6, naweza mara kuzidisha na kugawanya kazi za busara. Hakuna kitu katika nguzo nyingine.

    b Kama wengi wa hundi yako walikuwa:

    ... kwa ujasiri. Hongera! Umefanikiwa malengo yako katika sehemu hii! Fikiria ujuzi wa kujifunza uliyotumia ili uweze kuendelea kuitumia. Ulifanya nini ili uwe na ujasiri wa uwezo wako wa kufanya mambo haya? Kuwa maalum!

    ... kwa msaada fulani. Hii lazima kushughulikiwa haraka kama mada wewe si bwana kuwa mashimo katika barabara yako ya mafanikio. Math ni mtiririko - kila mada hujenga juu ya kazi ya awali. Ni muhimu kuhakikisha kuwa na msingi imara kabla ya kuendelea. Nani unaweza kuomba msaada? Washiriki wenzako na mwalimu ni rasilimali nzuri. Je, kuna mahali kwenye chuo ambapo waalimu hisabati zinapatikana? Je, ujuzi wako wa kujifunza unaweza kuboreshwa?

    ... hapana - Siipati! Hii ni muhimu na haipaswi kupuuza. Unahitaji kupata msaada mara moja au utazidiwa haraka. Angalia mwalimu wako haraka iwezekanavyo ili kujadili hali yako. Pamoja unaweza kuja na mpango wa kupata msaada unayohitaji.