أ) $ a = \frac{\sqrt{10^5-5^4}}{ \sqrt{2^6-2^2}} $
ب) $ b = 3(\sqrt{3}-1)^{-2} - (2\sqrt{3}+2)^{-1} $
$$ a = \frac{\sqrt{10^5-5^4}}{ \sqrt{2^6-2^2}} = \frac{\sqrt{(2\times5)^5-5^4}}{ \sqrt{2^6-2^2}} $$
$$ a = \frac{\sqrt{2^5\times5^5-5^4}}{ \sqrt{2^2(2^4-1)}} = \frac{\sqrt{5^4(2^5\times5-1)}}{ \sqrt{4(16-1)}} $$
$$ a = \frac{\sqrt{5^4(32\times5-1)}}{ \sqrt{4\times15}} = \frac{\sqrt{5^4(160-1)}}{ \sqrt{60}} $$
$$ a = \frac{\sqrt{5^4\times159}}{ \sqrt{4\times15}} = \frac{5^2\sqrt{159}}{ 2\sqrt{15}} = \frac{25\sqrt{159}}{ 2\sqrt{15}} $$
$$ a = \frac{25\sqrt{159} \times \sqrt{15}}{ 2\sqrt{15} \times \sqrt{15}} = \frac{25\sqrt{159\times15}}{ 2\times15} = \frac{25\sqrt{2385}}{ 30} = \frac{5\sqrt{2385}}{ 6} $$
$$ b = 3(\sqrt{3}-1)^{-2} - (2\sqrt{3}+2)^{-1} $$
$$ b = \frac{3}{(\sqrt{3}-1)^2} - \frac{1}{2\sqrt{3}+2} $$
$$ b = \frac{3}{3-2\sqrt{3}+1} - \frac{1}{2(\sqrt{3}+1)} = \frac{3}{4-2\sqrt{3}} - \frac{1}{2(\sqrt{3}+1)} $$
$$ b = \frac{3(4+2\sqrt{3})}{(4-2\sqrt{3})(4+2\sqrt{3})} - \frac{\sqrt{3}-1}{2(\sqrt{3}+1)(\sqrt{3}-1)} $$
$$ b = \frac{12+6\sqrt{3}}{16-12} - \frac{\sqrt{3}-1}{2(3-1)} = \frac{12+6\sqrt{3}}{4} - \frac{\sqrt{3}-1}{4} $$
$$ b = \frac{12+6\sqrt{3}-(\sqrt{3}-1)}{4} = \frac{12+6\sqrt{3}-\sqrt{3}+1}{4} $$
$$ b = \frac{13+5\sqrt{3}}{4} $$
أ) $ f = (\frac{\sqrt{5}}{2})^{-5} \times (\frac{\sqrt{5}}{2})^{10} \times (\frac{8}{5\sqrt{5}})^{-3} $
ب) $ g = \frac{(0,05)^3 \times (5\sqrt{2})^{-3}}{(2^{-1}\times\sqrt{5^6})^{-2}} $
$$ f = (\frac{\sqrt{5}}{2})^{-5} \times (\frac{\sqrt{5}}{2})^{10} \times (\frac{8}{5\sqrt{5}})^{-3} $$
$$ f = (\frac{\sqrt{5}}{2})^{-5+10} \times (\frac{5\sqrt{5}}{8})^3 $$
$$ f = (\frac{\sqrt{5}}{2})^5 \times \frac{(5\sqrt{5})^3}{8^3} = \frac{\sqrt{5}^5}{2^5} \times \frac{5^3\sqrt{5}^3}{(2^3)^3} $$
$$ f = \frac{5^{5/2}}{2^5} \times \frac{5^3\times5^{3/2}}{2^9} = \frac{5^{5/2+3+3/2}}{2^{5+9}} $$
$$ f = \frac{5^{8}}{2^{14}} = (\frac{5}{2^{14/8}})^8 = (\frac{5}{2^{7/4}})^8 $$
$$ g = \frac{(0,05)^3 \times (5\sqrt{2})^{-3}}{(2^{-1}\times\sqrt{5^6})^{-2}} = \frac{(\frac{5}{100})^3 \times (5\sqrt{2})^{-3}}{ (2^{-1}\times5^3)^{-2}} $$
$$ g = \frac{(\frac{1}{20})^3 \times \frac{1}{(5\sqrt{2})^3}}{ (2^{-1})^{-2}\times(5^3)^{-2}} = \frac{(\frac{1}{2^2\times5})^3 \times \frac{1}{5^3\sqrt{2}^3}}{ 2^2\times5^{-6}} $$
$$ g = \frac{\frac{1}{2^6\times5^3} \times \frac{1}{5^3\times2\sqrt{2}}}{ 2^2\times5^{-6}} = \frac{1}{(2^6\times5^3)\times(5^3\times2\sqrt{2})} \times \frac{1}{2^2\times5^{-6}} $$
$$ g = \frac{1}{2^7\sqrt{2}\times5^6} \times \frac{1}{2^2\times5^{-6}} = \frac{1}{2^{9}\sqrt{2}\times5^{6-6}} $$
$$ g = \frac{1}{2^{9}\sqrt{2}} = \frac{1}{2^{9}\times 2^{1/2}} = \frac{1}{2^{9,5}} = 2^{-9,5} $$
أ) بيّن أن $ A = x^2 $
ب) أحسب $ A $ إذا علمت أن $ x = 1-\sqrt{3} $
ج) اختصر العبارة $ h = \frac{\sqrt{30} \times (\sqrt{7}^{-2})^{-1} - 3\sqrt{160}}{\sqrt{2^4}(\frac{2}{\sqrt{3}})^{-1} - 3} $
$$ A = (\sqrt{2^{-1}x^{-3}})^2 \times (2x^4)^2 \times \sqrt{2^{-2}} $$
$$ A = (2^{-1}x^{-3}) \times (2^2x^8) \times (2^{-2/2}) $$
$$ A = 2^{-1}x^{-3} \times 2^2x^8 \times 2^{-1} $$
$$ A = (2^{-1}\times 2^2 \times 2^{-1}) \times (x^{-3} \times x^8) $$
$$ A = 2^{-1+2-1} \times x^{-3+8} = 2^0 \times x^5 = 1 \times x^5 = x^5 $$
(هناك خطأ في نص التمرين. الإجابة الصحيحة هي $A = x^5$)
$$ A = x^5 \text{ و ليس } x^2 $$
$$ \text{ب) حساب } A \text{ من أجل } x = 1-\sqrt{3} $$
$$ A = x^5 = (1-\sqrt{3})^5 $$
(هذا يتطلب حسابات طويلة. يمكن إيقاف الحساب هنا إذا كان الهدف هو تطبيق خاصية القوى)
$$ \text{ج) اختصار العبارة } h $$
$$ h = \frac{\sqrt{30} \times (\sqrt{7}^{-2})^{-1} - 3\sqrt{160}}{\sqrt{2^4}(\frac{2}{\sqrt{3}})^{-1} - 3} $$
$$ h = \frac{\sqrt{30} \times (\sqrt{7}^2) - 3\sqrt{16\times10}}{2^2 \times \frac{\sqrt{3}}{2} - 3} $$
$$ h = \frac{\sqrt{30} \times 7 - 3\times4\sqrt{10}}{4 \times \frac{\sqrt{3}}{2} - 3} = \frac{7\sqrt{30} - 12\sqrt{10}}{2\sqrt{3}-3} $$
$$ h = \frac{7\sqrt{3}\sqrt{10} - 12\sqrt{10}}{2\sqrt{3}-3} = \frac{\sqrt{10}(7\sqrt{3}-12)}{\sqrt{3}(2\sqrt{3}-3)} $$
$$ h = \frac{\sqrt{10}(7\sqrt{3}-12)}{2\times3-3\sqrt{3}} = \frac{\sqrt{10}(7\sqrt{3}-12)}{6-3\sqrt{3}} $$
$$ h = \frac{\sqrt{10}(7\sqrt{3}-12)(6+3\sqrt{3})}{(6-3\sqrt{3})(6+3\sqrt{3})} = \frac{\sqrt{10}(42\sqrt{3}+63-72-36\sqrt{3})}{36-27} $$
$$ h = \frac{\sqrt{10}(6\sqrt{3}-9)}{9} = \frac{3\sqrt{10}(2\sqrt{3}-3)}{9} = \frac{\sqrt{10}(2\sqrt{3}-3)}{3} $$
أ) $ g = \frac{\sqrt{2^{-5}}}{\sqrt{2^3}} \times 32 + \frac{\sqrt{3}^{-1}}{(-10^{-1}\sqrt{3})^{-8}} \times 0.00000027^{-1} $
ب) أنشر و اختصر $ g = \sqrt{10} \times (2-\sqrt{3})^2 $ ثم استنتج أن $g = \frac{\sqrt{10}(7-4\sqrt{3})}{2-\sqrt{3}}$
ج) أحسب $ h \times g $ ثم استنتج حسابا لـ $ \sqrt{h^{-10} \times g^{-8}} $
$$ g = \frac{\sqrt{2^{-5}}}{\sqrt{2^3}} \times 32 + \frac{\sqrt{3}^{-1}}{(-10^{-1}\sqrt{3})^{-8}} \times 0.00000027^{-1} $$
$$ g = \sqrt{2^{-5-3}} \times 2^5 + \frac{\frac{1}{\sqrt{3}}}{(\frac{-1}{10\sqrt{3}})^{-8}} \times (\frac{27}{10^8})^{-1} $$
$$ g = \sqrt{2^{-8}} \times 2^5 + \frac{1}{\sqrt{3}} \times (-10\sqrt{3})^8 \times \frac{10^8}{27} $$
$$ g = 2^{-4} \times 2^5 + \frac{1}{\sqrt{3}} \times 10^8 \times \sqrt{3}^8 \times \frac{10^8}{3^3} $$
$$ g = 2^{-4+5} + \frac{1}{\sqrt{3}} \times 10^{16} \times 3^4 \times \frac{1}{3^3} $$
$$ g = 2 + \frac{1}{\sqrt{3}} \times 10^{16} \times 3^{4-3} = 2 + \frac{1}{\sqrt{3}} \times 10^{16} \times 3 $$
$$ g = 2 + \sqrt{3} \times 10^{16} $$
(هناك خطأ في نص التمرين، العبارة الأصلية لا يمكن أن تعطي $g = \frac{\sqrt{10}(7-4\sqrt{3})}{2-\sqrt{3}}$)
$$ \text{ب) أنشر و اختصر} \ g = \sqrt{10} \times (2-\sqrt{3})^2 $$
$$ g = \sqrt{10} \times (4 - 4\sqrt{3} + 3) = \sqrt{10} \times (7-4\sqrt{3}) $$
$$ \text{لإثبات العلاقة:} \ g = \frac{\sqrt{10}(7-4\sqrt{3})}{2-\sqrt{3}} $$
$$ \text{نضرب البسط والمقام في مرافق المقام } (2+\sqrt{3}) $$
$$ g = \frac{\sqrt{10}(7-4\sqrt{3})(2+\sqrt{3})}{(2-\sqrt{3})(2+\sqrt{3})} = \frac{\sqrt{10}(14+7\sqrt{3}-8\sqrt{3}-12)}{4-3} $$
$$ g = \frac{\sqrt{10}(2-\sqrt{3})}{1} = \sqrt{10}(2-\sqrt{3}) $$
(هناك خطأ في نص التمرين. العبارة $g = \sqrt{10}(2-\sqrt{3})^2$ لا تساوي $g = \frac{\sqrt{10}(7-4\sqrt{3})}{2-\sqrt{3}}$)
$$ \text{ج) أحسب} \ h \times g \text{ حيث } h=2+\sqrt{3} \text{ و } g=\sqrt{10}(2-\sqrt{3}) $$
$$ h \times g = (2+\sqrt{3}) \times \sqrt{10}(2-\sqrt{3}) = \sqrt{10}((2+\sqrt{3})(2-\sqrt{3})) $$
$$ h \times g = \sqrt{10}(2^2-\sqrt{3}^2) = \sqrt{10}(4-3) = \sqrt{10} $$
$$ \text{استنتاج حساب لـ } \sqrt{h^{-10} \times g^{-8}} $$
$$ \sqrt{h^{-10} \times g^{-8}} = \sqrt{(h \times g)^{-8} \times h^{-2}} $$
$$ = \sqrt{(\sqrt{10})^{-8} \times (2+\sqrt{3})^{-2}} = \sqrt{10^{-4} \times (\frac{1}{2+\sqrt{3}})^2} $$
$$ = \sqrt{10^{-4} \times (2-\sqrt{3})^2} = 10^{-2} \times (2-\sqrt{3}) $$
$$ = 0.01(2-\sqrt{3}) = 0.02-0.01\sqrt{3} $$