أ) $ a = (-0,02)^{-2} \times 10^4 \times 10^{-3} \times (\frac{5}{100^{-1}})^{-2} $
ب) $ b = (1^{-2022} - 2\sqrt{3})^0 - (-3)^2 $
$$ a = (-0,02)^{-2} \times 10^4 \times 10^{-3} \times (\frac{5}{100^{-1}})^{-2} $$
$$ a = (-\frac{2}{100})^{-2} \times 10^{4-3} \times (5 \times 100)^{-2} $$
$$ a = (-\frac{1}{50})^{-2} \times 10^{1} \times (500)^{-2} $$
$$ a = (-50)^2 \times 10 \times (5 \times 10^2)^{-2} $$
$$ a = 2500 \times 10 \times 5^{-2} \times (10^2)^{-2} $$
$$ a = (25 \times 100) \times 10 \times \frac{1}{25} \times 10^{-4} $$
$$ a = (25 \times 10^2) \times 10^1 \times \frac{1}{25} \times 10^{-4} $$
$$ a = (25 \times \frac{1}{25}) \times 10^{2+1-4} $$
$$ a = 1 \times 10^{-1} = 0,1 $$
$$ b = (1^{-2022} - 2\sqrt{3})^0 - (-3)^2 $$
$$ b = 1 - 9 $$
$$ b = -8 $$
أ) $ c = (10^{-3})^{-2} \times 0,01^{-3} $
ب) $ b = -3\sqrt{5}^0 + \sqrt{3}^3 \times 5^{-1}\sqrt{3} $
ج) $ a = -\sqrt{3}^3 + \sqrt{3} \times 5^{-1} $
د) $ d = (-\sqrt{3})^2 + 3 \times 5^{-1} $
ه) $ e = (\frac{\sqrt{7}}{5})^0 + (\frac{\sqrt{7}}{5})^{-1} \times \sqrt{7}^3 $
و) $ f = (\frac{10^{-3}}{\sqrt{5}})^{-2} \times \frac{5}{0,01^{-3}} $
$$ c = (10^{-3})^{-2} \times 0,01^{-3} = 10^{(-3) \times (-2)} \times (\frac{1}{100})^{-3} $$
$$ c = 10^6 \times (10^{-2})^{-3} = 10^6 \times 10^6 $$
$$ c = 10^{6+6} = 10^{12} $$
$$ b = -3\sqrt{5}^0 + \sqrt{3}^3 \times 5^{-1}\sqrt{3} = -3 \times 1 + \sqrt{3}^{3+1} \times \frac{1}{5} $$
$$ b = -3 + \sqrt{3}^4 \times \frac{1}{5} = -3 + (\sqrt{3}^2)^2 \times \frac{1}{5} $$
$$ b = -3 + 3^2 \times \frac{1}{5} = -3 + 9 \times \frac{1}{5} $$
$$ b = -3 + \frac{9}{5} = -\frac{15}{5} + \frac{9}{5} = -\frac{6}{5} $$
$$ a = -\sqrt{3}^3 + \sqrt{3} \times 5^{-1} = -\sqrt{3}^2 \times \sqrt{3} + \frac{\sqrt{3}}{5} $$
$$ a = -3\sqrt{3} + \frac{\sqrt{3}}{5} = \sqrt{3}(-3+\frac{1}{5}) $$
$$ a = \sqrt{3}(-\frac{15}{5}+\frac{1}{5}) = -\frac{14\sqrt{3}}{5} $$
$$ d = (-\sqrt{3})^2 + 3 \times 5^{-1} = 3 + \frac{3}{5} $$
$$ d = \frac{15}{5} + \frac{3}{5} = \frac{18}{5} $$
$$ e = (\frac{\sqrt{7}}{5})^0 + (\frac{\sqrt{7}}{5})^{-1} \times \sqrt{7}^3 = 1 + \frac{5}{\sqrt{7}} \times \sqrt{7}^2 \times \sqrt{7} $$
$$ e = 1 + \frac{5}{\sqrt{7}} \times 7 \times \sqrt{7} = 1 + 5 \times 7 $$
$$ e = 1 + 35 = 36 $$
$$ f = (\frac{10^{-3}}{\sqrt{5}})^{-2} \times \frac{5}{0,01^{-3}} = \frac{10^{(-3) \times (-2)}}{\sqrt{5}^{-2}} \times \frac{5}{(10^{-2})^{-3}} $$
$$ f = \frac{10^6}{\frac{1}{(\sqrt{5})^2}} \times \frac{5}{10^6} = \frac{10^6}{\frac{1}{5}} \times \frac{5}{10^6} $$
$$ f = (10^6 \times 5) \times (\frac{5}{10^6}) = 5 \times 5 $$
$$ f = 25 $$
أ) $ N = \frac{0,5^{-3} \times 2^{-9}}{(\frac{1}{2})^{-4}} $
ب) $ M = (\frac{2}{\sqrt{5}})^{-7} \times (\frac{5}{4})^{-4} $
ج) $ L = 25^{-3} \times (\frac{1}{5})^{-4} $
د) $ c = (\frac{3}{4})^{-5} \times (\frac{-\sqrt{3}}{2})^{10} $
ه) $ d = \frac{(\frac{\sqrt{7}}{3})^{-2} \times (\frac{9}{7})^{-5}}{(\frac{\sqrt{7}}{5})^8} $
$$ N = \frac{0,5^{-3} \times 2^{-9}}{(\frac{1}{2})^{-4}} = \frac{(\frac{1}{2})^{-3} \times 2^{-9}}{2^4} $$
$$ N = \frac{(2^{-1})^{-3} \times 2^{-9}}{2^4} = \frac{2^3 \times 2^{-9}}{2^4} $$
$$ N = \frac{2^{3-9}}{2^4} = \frac{2^{-6}}{2^4} $$
$$ N = 2^{-6-4} = 2^{-10} $$
$$ M = (\frac{2}{\sqrt{5}})^{-7} \times (\frac{5}{4})^{-4} = (\frac{\sqrt{5}}{2})^7 \times (\frac{4}{5})^4 $$
$$ M = \frac{\sqrt{5}^7}{2^7} \times \frac{(2^2)^4}{5^4} = \frac{\sqrt{5}^7}{2^7} \times \frac{2^8}{5^4} $$
$$ M = \frac{5^{7/2}}{2^7} \times \frac{2^8}{5^4} = 5^{7/2-4} \times 2^{8-7} $$
$$ M = 5^{7/2-8/2} \times 2^1 = 5^{-1/2} \times 2 $$
(هنا لا يمكن كتابتها في صيغة قوة واحدة)
$$ L = 25^{-3} \times (\frac{1}{5})^{-4} = (5^2)^{-3} \times (5^{-1})^{-4} $$
$$ L = 5^{-6} \times 5^4 = 5^{-6+4} $$
$$ L = 5^{-2} $$
$$ c = (\frac{3}{4})^{-5} \times (\frac{-\sqrt{3}}{2})^{10} = (\frac{3}{4})^{-5} \times ((\frac{-\sqrt{3}}{2})^2)^5 $$
$$ c = (\frac{3}{4})^{-5} \times (\frac{3}{4})^5 $$
$$ c = (\frac{3}{4})^{-5+5} = (\frac{3}{4})^0 $$
$$ d = \frac{(\frac{\sqrt{7}}{3})^{-2} \times (\frac{9}{7})^{-5}}{(\frac{\sqrt{7}}{5})^8} = \frac{(\frac{3}{\sqrt{7}})^2 \times (\frac{7}{9})^5}{(\frac{\sqrt{7}}{5})^8} $$
$$ d = \frac{\frac{3^2}{\sqrt{7}^2} \times \frac{7^5}{(3^2)^5}}{(\frac{\sqrt{7}}{5})^8} = \frac{\frac{9}{7} \times \frac{7^5}{3^{10}}}{\frac{\sqrt{7}^8}{5^8}} $$
$$ d = \frac{9 \times 7^{5-1}}{3^{10}} \times \frac{5^8}{7^4} = \frac{3^2 \times 7^4}{3^{10}} \times \frac{5^8}{7^4} $$
$$ d = 3^{2-10} \times 7^{4-4} \times 5^8 = 3^{-8} \times 7^0 \times 5^8 $$
$$ d = 3^{-8} \times 5^8 = (\frac{5}{3^{-1}})^8 = (15)^8 $$
أ) بيّن أن $ S_{ABC} = BC^2 \times \frac{\sqrt{3}}{4} $.
ب) أكتب $ S $ و $ \frac{P}{16} $ في صيغة قوة لعدد حقيقي دليلها مخالف لواحد.
$$ \text{لدينا} \ ABC \ \text{مثلث متقايس الأضلاع طول ضلعه} \ BC = a $$
$$ \text{أ) حساب المساحة} \ S_{ABC} $$
$$ S_{ABC} = \frac{\text{القاعدة} \times \text{الارتفاع}}{2} = \frac{BC \times AH}{2} $$
$$ \text{في مثلث متقايس الأضلاع، الارتفاع} \ AH = \frac{\sqrt{3}}{2} \times BC $$
$$ S_{ABC} = \frac{BC \times (\frac{\sqrt{3}}{2} \times BC)}{2} $$
$$ S_{ABC} = \frac{BC^2 \times \frac{\sqrt{3}}{2}}{2} $$
$$ S_{ABC} = BC^2 \times \frac{\sqrt{3}}{2} \times \frac{1}{2} = BC^2 \times \frac{\sqrt{3}}{4} $$
$$ \text{ب) كتابة} \ S \ \text{و} \ \frac{P}{16} \ \text{في صيغة قوة} $$
$$ S = BC^2 \times \frac{\sqrt{3}}{4} $$
$$ \text{بما أن} \ BC = a \ \text{فإن} \ S = a^2 \times \frac{\sqrt{3}}{4} $$
$$ \text{محيط المثلث} \ P = 3 \times BC = 3a $$
$$ \frac{P}{16} = \frac{3a}{16} $$
(هنا لا يمكن كتابته في صيغة قوة لعدد حقيقي دليلها مخالف لواحد)