5.3E: Mazoezi

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Mazoezi hufanya kamili

Kurahisisha Maneno Kutumia Mali kwa Watazamaji

Katika mazoezi yafuatayo, kurahisisha kila kujieleza kwa kutumia mali kwa exponents.

1. ⓐ$d^3·d^6$ ⓑ$4^{5x}·4^{9x}$ ⓒ$2y·4y^3$ ⓓ$w·w^2·w^3$

Jibu: ⓐ$d^9$ ⓑ$4^{14x}$ ⓒ$8y^4$ ⓓ$w^6$

2. ⓐ$x^4·x^2$ ⓑ$8^{9x}·8^3$ ⓒ$3z^{25}·5z^8$ ⓓ$y·y^3·y^5$

3. ⓐ$n^{19}·n^{12}$ ⓑ$3^x·3^6$ ⓒ$7w^5·8w$ ⓓ$a^4·a^3·a^9$

Jibu: ⓐ$n^{31}$ ⓑ$3^{x+6}$ ⓒ$56w^6$
ⓓ$a^{16}$

4. ⓐ$q^{27}·q^{15}$ ⓑ$5^x·5^{4x}$ ⓒ$9u^{41}·7u^{53}$
ⓓ$c^5·c^{11}·c^2$

5. $m^x·m^3$

Jibu: $m^{x+3}$

6. $n^y·n^2$

7. $y^a·y^b$

Jibu: $y^{a+b}$

8. $x^p·x^q$

9. ⓐ$\dfrac{x^{18}}{x^3}$ ⓑ$\dfrac{5^{12}}{5^3}$ ⓒ$\dfrac{q^{18}}{q^{36}}$ ⓓ$\dfrac{10^2}{10^3}$

Jibu: ⓐ$x^{15}$ ⓑ$5^9$ ⓒ$\dfrac{1}{q^{18}}$ ⓓ$\dfrac{1}{10}$

10. ⓐ$\dfrac{y^{20}}{y^{10}}$ ⓑ$\dfrac{7^{16}}{7^2}$ ⓒ$\dfrac{t^{10}}{t^{40}}$ ⓓ$\dfrac{8^3}{8^5}$

11. ⓐ$\dfrac{p^{21}}{p^7}$ ⓑ$\dfrac{4^{16}}{4^4}$ ⓒ$\dfrac{b}{b^9}$ ⓓ$\dfrac{4}{4^6}$

Jibu: ⓐ$p^{14}$ ⓑ$4^{12}$ ⓒ$\dfrac{1}{b^8}$ ⓓ$\dfrac{1}{4^5}$

12. ⓐ$\dfrac{u^{24}}{u^3}$ ⓑ$\dfrac{9^{15}}{9^5}$ ⓒ$\dfrac{x}{x^7}$ ⓓ$\dfrac{10}{10^3}$

13. ⓐ$20^0$ ⓑ$b^0$

Jibu: ⓐ 1 ⓑ 1

14. ⓐ$13^0$ ⓑ$k^0$

15. ⓐ$−27^0$ ⓑ$−(27^0)$

Jibu: ⓐ$−1$ ⓑ$−1$

16. ⓐ$−15^0$ ⓑ$−(15^0)$

Tumia Ufafanuzi wa Mtazamaji Mbaya

Katika mazoezi yafuatayo, kurahisisha kila kujieleza.

17. ⓐ$a^{−2}$ ⓑ$10^{−3}$ ⓒ$\dfrac{1}{c^{−5}}$ ⓓ$\dfrac{1}{3^{−2}}$

Jibu: ⓐ$\dfrac{1}{a^{2}}$ ⓑ$\dfrac{1}{1000}$ ⓒ$c^{5}$ ⓓ$9$

18. ⓐ$b^{−4}$ ⓑ$10^{−2}$ ⓒ$\dfrac{1}{c^{−5}}$ ⓓ$\dfrac{1}{5^{−2}}$

19. ⓐ$r^{−3}$ ⓑ$10^{−5}$ ⓒ$\dfrac{1}{q^{−10}}$ ⓓ$\dfrac{1}{10^{−3}}$

Jibu: ⓐ$\dfrac{1}{r3}$ ⓑ$\dfrac{1}{100,000}$ ⓒ$q^{10}$ ⓓ$1,000$

20. ⓐ$s^{−8}$ ⓑ$10^{−2}$ ⓒ$\dfrac{1}{t^{−9}}$ ⓓ$\dfrac{1}{10^{−4}}$

21. ⓐ$\left(\dfrac{5}{8}\right)^{-2}$ ⓑ$\left(−\dfrac{b}{a}\right)^{−2}$

Jibu: ⓐ$\dfrac{64}{25}$ ⓑ$\dfrac{a^{2}}{b^{2}}$

22. ⓐ$\left(\dfrac{3}{10}\right)^{−2}$ ⓑ$\left(−\dfrac{2}{z}\right)^{−3}$

23. ⓐ$\left(\dfrac{4}{9}\right)^{−3}$ ⓑ$\left(−\dfrac{u}{v}\right)^{−5}$

Jibu: ⓐ$\dfrac{729}{64}$ ⓑ$−\dfrac{v^{5}}{u^{5}}$

24. ⓐ$\left(\dfrac{7}{2}\right)^{−3}$ ⓑ$\left(−\dfrac{3}{x}\right)^{−3}$

25. ⓐ$(−5)^{−2}$ ⓑ$−5^{−2}$ ⓒ$\left(−\dfrac{1}{5}\right)^{−2}$ ⓓ$−\left(\dfrac{1}{5}\right)^{−2}$

Jibu: ⓐ$\dfrac{1}{25}$ ⓑ$−\dfrac{1}{25}$ ⓒ$25$ ⓓ$−25$

26. ⓐ$−5^{−3}$ ⓑ$\left(−\dfrac{1}{5}\right)^{−3}$ ⓒ$−\left(\dfrac{1}{5}\right)^{−3}$ ⓓ$(−5)^{−3}$

27. ⓐ$3·5^{−1}$ ⓑ$(3·5)^{−1}$

Jibu: ⓐ$\dfrac{3}{5}$ ⓑ$\dfrac{1}{15}$

28. ⓐ$3·4^{−2}$ ⓑ$(3·4)^{−2}$

Katika mazoezi yafuatayo, kurahisisha kila kujieleza kwa kutumia Mali ya Bidhaa.

29. ⓐ$b^{4}b^{−8}$ ⓑ$(w^{4}x^{−5})(w^{−2}x^{−4})$) ⓒ$(−6c^{−3}d^9)(2c^4d^{−5})$

Jibu: ⓐ$\dfrac{1}{b^{4}}$ ⓑ$\dfrac{w^{2}}{x^{9}}$ ⓒ$−12cd^{4}$

30. ⓐ$s^{3}·s^{−7}$ ⓑ$(m^{3}n^{−3})(m^{5}n^{−1})$
ⓒ$(−2j^{−5}k^{8})(7j^{2}k^{−3})$

31. ⓐ$a^{3}·a^{−3}$ ⓑ$(uv^{−2})(u^{−5}v^{−3})$
ⓒ$(−4r^{−2}s^{−8})(9r^{4}s^{3})$

Jibu: ⓐ$1$ ⓑ$\dfrac{1}{u^{4}v^{5}}$ ⓒ$−36\dfrac{r^{2}}{j^{5}}$

32. ⓐ$y^{5}·y^{−5}$ ⓑ$(pq^{−4})(p^{−6}q^{−3})$
ⓒ$(−5m^{4}n^{6})(8m^{−5}n^{−3})$

33. $p^{5}·p^{−2}·p^{−4}$

Jibu: $\dfrac{1}{p}$

34. $x^{4}·x^{−2}·x^{−3}$

Katika mazoezi yafuatayo, kurahisisha kila kujieleza kwa kutumia Power Mali.

35. ⓐ$(m^4)^2$ ⓑ$(10^3)^6$ ⓒ$(x^3)^{−4}$

Jibu: ⓐ$m^{8}$ ⓑ$10^{18}$ ⓒ$\dfrac{1}{x^{12}}$

36. ⓐ$(b^{2})^{7}$ ⓑ$(3^8)^2$ ⓒ$(k^2)^{−5}$

37. ⓐ$(y^3)^x$ ⓑ$(5^x)^x$ ⓒ$(q^6)^{−8}$

Jibu: ⓐ$y^{3x}$ ⓑ$5^{xy}$ ⓒ$\dfrac{1}{q^{48}}$

38. ⓐ$(x^2)^y$ ⓑ$(7^a)^b$ ⓒ$(a^9)^{−10}$

Katika mazoezi yafuatayo, kurahisisha kila kujieleza kwa kutumia Bidhaa kwa Power Mali.

39. ⓐ$(−3xy)^2$ ⓑ$(6a)^0$ ⓒ$(5x^2)^{−2}$ ⓓ$(−4y^{−3})^2$

Jibu: ⓐ$9x^2y^2$ ⓑ 1 ⓒ$\dfrac{1}{25x^4}$ ⓓ$\dfrac{16}{y^6}$

40. ⓐ$(−4ab)^2$ ⓑ$(5x)^0$ ⓒ$(4y^3)^{−3}$ ⓓ$(−7y^{−3})^2$

41. ⓐ$(−5ab)^3$ ⓑ$(−4pq)^0$ ⓒ$(−6x^3)^{−2}$ ⓓ$(3y^{−4})^2$

Jibu: ⓐ$−125a^3b^3$ ⓑ 1 ⓒ$\dfrac{1}{36x^6}$ ⓓ$\dfrac{9}{y^8}$

42. ⓐ$(−3xyz)^4$ ⓑ$(−7mn)^0$ ⓒ$(−3x^3)^{−2}$
ⓓ$(2y^{−5})^2$

Katika mazoezi yafuatayo, kurahisisha kila kujieleza kwa kutumia Quotient kwa Power Mali.

43. ⓐ$(p^2)^5$ ⓑ$\left(\dfrac{x}{y}\right)^{−6}$ ⓒ$\left(\dfrac{2xy^2}{z}\right)^3$ ⓓ$\left(\dfrac{4p^{−3}}{q^2}\right)^2$

Jibu: ⓐ$\dfrac{p^5}{32}$ ⓑ$\dfrac{y^6}{x^6}$ ⓒ$\dfrac{8x^3y^6}{z^3}$
ⓓ$\dfrac{16}{p^6q^4}$

44. ⓐ$\left(\dfrac{x}{3}\right)^4$ ⓑ$\left(\dfrac{a}{b}\right)^{−5}$ ⓒ$\left(\dfrac{2xy^2}{z}\right)^3$ ⓓ$\left(\dfrac{x^3y}{z^4}\right)^2$

45. ⓐ$\left(\dfrac{a}{3b}\right)^4$ ⓑ$\left(\dfrac{5}{4m}\right)^{−2}$ ⓒ$\left(\dfrac{3a^{−2}b^3}{c^3}\right)^{−2}$ ⓓ$\left(\dfrac{p^{−1}q^4}{r^{−4}}\right)^2$

Jibu: ⓐ$\dfrac{a^4}{81b^4}$ ⓑ$\dfrac{16m^2}{25}$ ⓒ$\dfrac{a^4c^4}{9b^6}$ ⓓ$\dfrac{q^8r^8}{p^2}$

46. ⓐ$\left(\dfrac{x^2}{y}\right)^3$ ⓑ$\left(\dfrac{10}{3q}\right)^{−4}$ ⓒ$\left(\dfrac{2x^3y^4}{3z^2}\right)^5$ ⓓ$\left(\dfrac{5a^3b^{−1}}{2c^4}\right)^{−3}$

Katika mazoezi yafuatayo, kurahisisha kila kujieleza kwa kutumia mali kadhaa.

47. ⓐ$(5t^2)^3(3t)^2$ ⓑ$\dfrac{(t^2)^5(t^{−4})^2}{(t^3)^7}$ ⓒ$\left(\dfrac{2xy^2}{x^3y^{−2}}\right)^2\left(\dfrac{12xy^3}{x^3y^{−1}}\right)^{−1}$

Jibu: ⓐ$1125t^8$ ⓑ$\dfrac{1}{t^{19}}$ ⓒ$\dfrac{y^4}{3x^2}$

48. ⓐ$(10k^4)^3(5k^6)^2$ ⓑ$\dfrac{(q^3)^6(q^{−2})^3}{(q^4)^8}$

49. ⓐ$(m^2n)^2(2mn^5)^4$ ⓑ$\dfrac{(−2p^{−2})^4(3p^4)^2}{(−6p^3)^2}$

Jibu: ⓐ$16m^8n^{22}$ ⓑ$\dfrac{4}{p^6}$

50. ⓐ$(3pq^4)^2(6p^6q)^2$ ⓑ$\dfrac{(−2k^{−3})^2(6k^2)^4}{(9k^4)^2}$

Mazoezi ya mchanganyiko

Katika mazoezi yafuatayo, kurahisisha kila kujieleza.

51. ⓐ$7n^{−1}$ ⓑ$(7n)^{−1}$ ⓒ$(−7n)^{−1}$

Jibu: ⓐ$\dfrac{7}{n}$ ⓑ$\dfrac{1}{7n}$ ⓒ$−\dfrac{1}{7n}$

52. ⓐ$6r^{−1}$ ⓑ$(6r)^{−1}$ ⓒ$(−6r)^{−1}$

53. ⓐ$(3p)^{−2}$ ⓑ$3p^{−2}$ ⓒ$−3p^{−2}$

Jibu: ⓐ$\dfrac{1}{9p^2}$ ⓑ$\dfrac{3}{p^2}$ ⓒ$−\dfrac{3}{p^2}$

54. ⓐ$(2q)^{−4}$ ⓑ$2q^{−4}$ ⓒ$−2q^{−4}$

55. $(x^2)^4·(x^3)^2$

Jibu: $x^{14}$

56. $(y^4)^3·(y^5)^2$

57. $(a^2)^6·(a^3)^8$

Jibu: $a^{30}$

58. $(b^7)^5·(b^2)^6$

59. $(2m^6)^3$

Jibu: $2m^{18}$

60. $(3y^2)^4$

61. $(10x^2y)^3$

Jibu: $1,000x^6y^3$

62. $(2mn^4)^5$

63. $(−2a^3b^2)^4$

Jibu: $16a^{12}b^8$

64. $(−10u^2v^4)^3$

65. $\left(\dfrac{2}{3}x^2y\right)^3$

Jibu: $\dfrac{8}{27}x^6y^3$

66. $\left(\dfrac{7}{9}pq^4\right)^2$

67. $(8a^3)^2(2a)^4$

Jibu: $1,024a^{10}$

68. $(5r^2)^3(3r)^2$

69. $(10p^4)^3(5p^6)^2$

Jibu: $25,000p^{24}$

70. $(4x^3)^3(2x^5)^4$

71. $\left(\dfrac{1}{2}x^2y^3\right)^4\left(4x^5y^3\right)^2$

Jibu: $x^{18}y^{18}$

72. $\left(\dfrac{1}{3}m^3n^2\right)^4\left(9m^8n^3\right)^2$

73. $(3m^2n)^2(2mn^5)^4$

Jibu: $144m^8n^{22}$

74. $(2pq^4)^3(5p^6q)^2$

75. ⓐ$(3x)^2(5x)$ ⓑ$(2y)^3(6y)$

Jibu: ⓐ$45x^3$ ⓑ$48y^4$

76. ⓐ$\left(\dfrac{1}{2}y^2\right)^3\left(\dfrac{2}{3}y\right)^2$ ⓑ$\left(\dfrac{1}{2}j^2\right)^5\left(\dfrac{2}{5}j^3\right)^2$

77. ⓐ$(2r^{−2})^3(4^{−1}r)^2$ ⓑ$(3x^{−3})^3(3^{−1}x^5)^4$

Jibu: ⓐ$12r^4$ ⓑ$13x^{11}$

78. $\left(\dfrac{k^{−2}k^8}{k^3}\right)^2$

79. $\left(\dfrac{j^{−2}j^5}{j^4}\right)^3$

Jibu: $\dfrac{1}{j^3}$

80. $\dfrac{(−4m^{−3})^2(5m^4)^3}{(−10m^6)^3}$

81. $\dfrac{(−10n^{−2})^3(4n^5)^2}{(2n^8)^2}$

Jibu: $−\dfrac{4000}{n^{12}}$

Tumia Nukuu ya kisayansi

Katika mazoezi yafuatayo, weka kila nambari katika maelezo ya kisayansi.

82. ⓐ 57,000 ⓑ 0.026

83. ⓐ 340,000 ⓑ 0.041

Jibu: ⓐ$34\times10^4$ ⓑ$41\times10^{−3}$

84. ⓐ 8,750,000 ⓑ 0.00000871

85. ⓐ 1,290,000 ⓑ 0.00000103

Jibu

ⓐ$1.29\times10^6$

ⓑ$103\times10^{−8}$

Katika mazoezi yafuatayo, kubadilisha kila nambari kwa fomu ya decimal.

86. ⓐ$5.2\times10^2$ ⓑ$2.5\times10^{−2}$

87. ⓐ$−8.3\times10^2$ ⓑ$3.8\times10^{−2}$

Jibu: ⓐ$−830$ ⓑ 0.038

88. ⓐ$7.5\times10^6$ ⓑ$−4.13\times10^{−5}$

89. ⓐ$1.6\times10^{10}$ ⓑ$8.43\times10^{−6}$

Jibu: ⓐ 16,000,000,000
ⓑ 0.00000843

Katika mazoezi yafuatayo, kuzidisha au kugawanya kama ilivyoonyeshwa. Andika jibu lako kwa fomu ya decimal.

90. ⓐ$(3\times10^{−5})(3\times10^9)$ ⓑ$\dfrac{7\times10^{−3}}{1\times10^{−7}}$

91. ⓐ$(2\times10^2)(1\times10^{−4})$ ⓑ$\dfrac{5\times10^{−2}}{1\times10^{−10}}$

Jibu: ⓐ 0.02 ⓑ 500,000,000

92. ⓐ$(7.1\times10^{−2})(2.4\times10^{−4})$ ⓑ$\dfrac{6\times10^4}{3\times10^{−2}}$

93. ⓐ$(3.5\times10^{−4})(1.6\times10^{−2})$ ⓑ$\dfrac{8\times10^6}{4\times10^{−1}}$

Jibu: ⓐ 0.0000056 ⓑ 20,000,000

Mazoezi ya kuandika

94. Matumizi ya Bidhaa Mali kwa ajili ya Exponents kueleza kwa nini$x·x=x^2$.

95. Jennifer anadhani quotient$\dfrac{a^{24}}{a^6}$ simplifies kwa$a^4$. Ni nini kibaya na mawazo yake?

Jibu: Majibu yatatofautiana.

96. Eleza kwa nini$−5^3=(−5)^3$ lakini$−5^4 \neq (−5)^4$.

97. Unapobadilisha nambari kutoka kwa nukuu ya decimal hadi nukuu ya kisayansi, unajuaje kama mtangazaji atakuwa chanya au hasi?

Jibu: Majibu yatatofautiana.

Self Check

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

$Jedwali hili lina safu 4 na nguzo 4. Mstari wa kwanza ni mstari wa kichwa na huandika kila safu. Kichwa cha kwanza cha safu ni “Naweza...”, pili ni “Kwa uaminifu”, ya tatu ni “Kwa msaada fulani”, na ya nne ni “Hapana, siipati”. Chini ya safu ya kwanza ni maneno “kurahisisha maneno kwa kutumia mali kwa vielelezo.”, “tumia ufafanuzi wa exponent hasi”, na “tumia nukuu ya kisayansi”. Nguzo nyingine zimeachwa tupu ili mwanafunzi aweze kuonyesha kiwango chao cha ustadi kwa kila mada.$

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