تمارين حول قوى الأعداد الكسرية

7

اكتب في صيغة قوة لعدد كسري

  • $ A = \left(\frac{3}{5}\right)^{-12} \times \left(\frac{3}{5}\right)^{18} $
  • $ B = \left(-\frac{1}{2}\right)^{18} \times \left(\frac{1}{2}\right)^{19} $
  • $ C = \left[\left(\frac{1}{2}\right)^3\right]^{-2} $
  • $ D = \left(\frac{11}{5}\right)^{-17} \times \frac{11}{5} $
  • $ E = \left(\frac{4}{3}\right)^{18} \times \left(\frac{9}{16}\right)^{-19} $
  • $ F = \left(5^4 \times 3^2\right)^4 \times 3^{24} $

الحل المفصل للتمرين 7

$ A = \left(\frac{3}{5}\right)^{-12} \times \left(\frac{3}{5}\right)^{18} = \left(\frac{3}{5}\right)^{-12+18} = \left(\frac{3}{5}\right)^6 $

$ B = \left(-\frac{1}{2}\right)^{18} \times \left(\frac{1}{2}\right)^{19} = \left(\frac{1}{2}\right)^{18} \times \left(\frac{1}{2}\right)^{19} = \left(\frac{1}{2}\right)^{18+19} = \left(\frac{1}{2}\right)^{37} $ (لأن دليل القوة 18 زوجي)

$ C = \left[\left(\frac{1}{2}\right)^3\right]^{-2} = \left(\frac{1}{2}\right)^{3 \times (-2)} = \left(\frac{1}{2}\right)^{-6} $

$ D = \left(\frac{11}{5}\right)^{-17} \times \left(\frac{11}{5}\right)^1 = \left(\frac{11}{5}\right)^{-17+1} = \left(\frac{11}{5}\right)^{-16} $

$ E = \left(\frac{4}{3}\right)^{18} \times \left(\left(\frac{3}{4}\right)^2\right)^{-19} = \left(\frac{4}{3}\right)^{18} \times \left(\frac{3}{4}\right)^{-38} = \left(\frac{4}{3}\right)^{18} \times \left(\frac{4}{3}\right)^{38} = \left(\frac{4}{3}\right)^{18+38} = \left(\frac{4}{3}\right)^{56} $

$ F = (5^4 \times 3^2)^4 \times 3^{24} = (5^{4 \times 4} \times 3^{2 \times 4}) \times 3^{24} = (5^{16} \times 3^8) \times 3^{24} = 5^{16} \times 3^{8+24} = 5^{16} \times 3^{32} = 5^{16} \times (3^2)^{16} = 5^{16} \times 9^{16} = (5 \times 9)^{16} = 45^{16} $

8

اكتب في صيغة قوة لعدد كسري

  • $ A = \left(\frac{9}{5}\right)^{-11} \times \left(\frac{9}{5}\right)^{20} $
  • $ B = \left(\frac{7}{5}\right)^{29} \times \left(\frac{7}{5}\right)^{-8} $
  • $ C = \frac{\left(\frac{7}{3}\right)^{24}}{\left(\frac{7}{3}\right)^{14}} $
  • $ D = \left(-\frac{3}{5}\right)^2 \times \left(\frac{3}{2}\right)^3 \div \left(\frac{1}{2}\right)^5 $
  • $ E = \left(\frac{27}{8}\right)^{-2} \times \left(\frac{3}{2}\right)^4 $
  • $ F = \frac{0.001^4 \times 1000^{-2}}{100^3 \times 0.01^5} $

الحل المفصل للتمرين 8

$ A = \left(\frac{9}{5}\right)^{-11 + 20} = \left(\frac{9}{5}\right)^9 $

$ B = \left(\frac{7}{5}\right)^{29 + (-8)} = \left(\frac{7}{5}\right)^{21} $

$ C = \left(\frac{7}{3}\right)^{24 - 14} = \left(\frac{7}{3}\right)^{10} $

$ D = \left(\frac{3}{5}\right)^2 \times \left(\frac{3}{2}\right)^3 \times \left(\frac{2}{1}\right)^5 = \frac{3^2}{5^2} \times \frac{3^3}{2^3} \times 2^5 = \frac{3^{2+3} \times 2^5}{5^2 \times 2^3} = \frac{3^5 \times 2^{5-3}}{5^2} = \frac{3^5 \times 2^2}{5^2} = 3^5 \times \left(\frac{2}{5}\right)^2 $ (لا يمكن تبسيطها أكثر في صيغة قوة واحدة)

$ E = \left(\left(\frac{3}{2}\right)^3\right)^{-2} \times \left(\frac{3}{2}\right)^4 = \left(\frac{3}{2}\right)^{-6} \times \left(\frac{3}{2}\right)^4 = \left(\frac{3}{2}\right)^{-6+4} = \left(\frac{3}{2}\right)^{-2} $

$ F = \frac{(10^{-3})^4 \times (10^3)^{-2}}{(10^2)^3 \times (10^{-2})^5} = \frac{10^{-12} \times 10^{-6}}{10^6 \times 10^{-10}} = \frac{10^{-18}}{10^{-4}} = 10^{-18 - (-4)} = 10^{-14} $

9

اكتب في صيغة قوة لعدد كسري

  • $ A = \left(\left(\frac{5}{7}\right)^{-120}\right)^4 $
  • $ B = (-0.9)^{-2009} \times \left(\frac{307}{333}\right)^{-2009} $
  • $ C = (-6.17)^{147} \times (-6.17)^{23} $
  • $ D = -(-2)^5 $
  • $ E = \left(\frac{-3.14}{3.14}\right)^{-219} \times \frac{1}{(-3)^{-343}} $
  • $ F = \frac{(-19)^{167}}{(-19)^{57}} $

الحل المفصل للتمرين 9

$ A = \left(\frac{5}{7}\right)^{-120 \times 4} = \left(\frac{5}{7}\right)^{-480} $

$ B = \left(-0.9 \times \frac{307}{333}\right)^{-2009} = \left(-\frac{9}{10} \times \frac{307}{333}\right)^{-2009} = \left(-\frac{307}{370}\right)^{-2009} $ (التبسيط ليس ضرورياً، القاعدة هي الأهم)

$ C = (-6.17)^{147+23} = (-6.17)^{170} $

$ D = -(-32) = 32 = 2^5 $

$ E = (-1)^{-219} \times (-3)^{343} = (-1) \times (-3)^{343} = -(-3^{343}) = 3^{343} $

$ F = (-19)^{167-57} = (-19)^{110} $

10

احسب العبارات التالية

  • $ \left(\frac{3}{5}\right)^{-2} = \ldots $
  • $ \left(\frac{-1}{2}\right)^4 = \ldots $
  • $ \left(\frac{20}{17}\right)^{2017} \times \left(\frac{20}{17}\right)^{-2016} = \ldots $
  • $ \left[\left(\frac{1}{2}\right)^3\right]^2 = \ldots $

الحل المفصل للتمرين 10

$ \left(\frac{3}{5}\right)^{-2} = \left(\frac{5}{3}\right)^2 = \frac{5^2}{3^2} = \frac{25}{9} $

$ \left(\frac{-1}{2}\right)^4 = \frac{(-1)^4}{2^4} = \frac{1}{16} $

$ \left(\frac{20}{17}\right)^{2017 - 2016} = \left(\frac{20}{17}\right)^1 = \frac{20}{17} $

$ \left(\frac{1}{2}\right)^{3 \times 2} = \left(\frac{1}{2}\right)^6 = \frac{1^6}{2^6} = \frac{1}{64} $