تمارين في الحساب على القوى والجذور

$$a = 5^{-1} \times \sqrt{2^2} - \sqrt{3^2}$$ $$a = \frac{1}{5} \times 2 - 3$$ $$a = \frac{2}{5} - \frac{15}{5}$$ $$a = \frac{2 - 15}{5} = \frac{-13}{5}$$

$$b = \frac{1}{(\sqrt{5}-2)^{-2012}} \times (\sqrt{5}+2)^{2012}$$ $$b = (\sqrt{5}-2)^{2012} \times (\sqrt{5}+2)^{2012}$$ $$b = ((\sqrt{5}-2)(\sqrt{5}+2))^{2012}$$ $$b = ((\sqrt{5})^2 - 2^2)^{2012}$$ $$b = (5 - 4)^{2012}$$ $$b = 1^{2012} = 1$$

$$c = \frac{2\sqrt{2}-1}{2\sqrt{2}+1} + \frac{2\sqrt{2}+1}{2\sqrt{2}-1}$$ $$c = \frac{(2\sqrt{2}-1)^2 + (2\sqrt{2}+1)^2}{(2\sqrt{2}+1)(2\sqrt{2}-1)}$$ $$c = \frac{(8 - 4\sqrt{2} + 1) + (8 + 4\sqrt{2} + 1)}{(2\sqrt{2})^2 - 1^2}$$ $$c = \frac{9 - 4\sqrt{2} + 9 + 4\sqrt{2}}{8 - 1}$$ $$c = \frac{18}{7}$$

$$(\frac{\sqrt{2}}{3})^{-7} \times (\frac{2}{9})^{2} = (\frac{2^{1/2}}{3})^{-7} \times (\frac{2}{3^2})^{2}$$ $$= (\frac{3}{2^{1/2}})^{7} \times \frac{2^2}{3^4}$$ $$= \frac{3^7}{2^{7/2}} \times \frac{2^2}{3^4} = \frac{3^3}{2^{7/2 - 2}} = \frac{3^3}{2^{3/2}}$$ $$= 3^3 \times 2^{-3/2} = (\frac{3}{2^{1/2}})^{3} = (\frac{3}{\sqrt{2}})^3$$

$$\frac{0.00001^{3} \times \frac{3}{100}}{0.0003 \times 10^{10}} = \frac{(10^{-5})^3 \times 3 \times 10^{-2}}{3 \times 10^{-4} \times 10^{10}}$$ $$= \frac{10^{-15} \times 3 \times 10^{-2}}{3 \times 10^{6}}$$ $$= \frac{3 \times 10^{-17}}{3 \times 10^{6}} = 10^{-17 - 6} = 10^{-23}$$

$$\frac{-5\sqrt{5} \times \sqrt{5}^{4}}{25^3} = \frac{-5^1 \times 5^{1/2} \times 5^{4/2}}{(5^2)^3}$$ $$= \frac{-5^{1+1/2+2}}{5^6} = \frac{-5^{7/2}}{5^6}$$ $$= -5^{7/2 - 6} = -5^{-5/2} = -(\frac{1}{5^{5/2}}) = -(\frac{1}{5^2 \sqrt{5}}) = -(\frac{\sqrt{5}}{125})$$

$$F = \frac{(a^{-2}b)^{-3} a^{-3}b^2}{a^2b^{-5}(ab^{-6})^{-1}}$$ $$= \frac{a^{-2 \times -3} b^{-3} a^{-3} b^2}{a^2 b^{-5} a^{-1} b^{-6 \times -1}}$$ $$= \frac{a^6 b^{-3} a^{-3} b^2}{a^2 b^{-5} a^{-1} b^{6}}$$ $$= \frac{a^{6-3} b^{-3+2}}{a^{2-1} b^{-5+6}}$$ $$= \frac{a^3 b^{-1}}{a^1 b^1}$$ $$= a^{3-1} b^{-1-1}$$ $$= a^2 b^{-2}$$

وهو المطلوب.

$$F = a^2 b^{-2} = \frac{a^2}{b^2} = (\frac{a}{b})^2$$ $$F = (3\sqrt{2} - 5)^2$$ $$F = (3\sqrt{2})^2 - 2(3\sqrt{2})(5) + 5^2$$ $$F = 9 \times 2 - 30\sqrt{2} + 25$$ $$F = 18 - 30\sqrt{2} + 25$$ $$F = 43 - 30\sqrt{2}$$

$$F = a^2 b^{-2}$$ $$F = a^2 a^{-2}$$ $$F = a^{2-2} = a^0 = 1$$

$$A = \frac{(3^{2} a^{-3}b^{-6})^{-1} \times 3^3 (a^1b^{-2})^{-5}}{(3a^{-2}b^{3})^{-4}}$$ $$= \frac{3^{-2}a^{3}b^{6} \times 3^3 a^{-5}b^{10}}{3^{-4}a^8b^{-12}}$$ $$= \frac{3^{-2+3} a^{3-5} b^{6+10}}{3^{-4}a^8b^{-12}}$$ $$= \frac{3^1 a^{-2} b^{16}}{3^{-4} a^8 b^{-12}}$$ $$= 3^{1-(-4)} a^{-2-8} b^{16-(-12)}$$ $$= 3^5 a^{-10} b^{28}$$

**ملاحظة**: يوجد خطأ في نص التمرين الأصلي. النتيجة الصحيحة للتبسيط هي $A = 3^5 a^{-10} b^{28}$، وليست $A=3^{-1}a^{8}b^{-4}$. سنكمل الحل على أساس النتيجة الصحيحة.

$$\frac{a}{b} = 3 \implies b = \frac{a}{3} = \frac{\sqrt{7}}{3}$$ $$A = 3^5 a^{-10} b^{28} = 3^5 \times (\sqrt{7})^{-10} \times (\frac{\sqrt{7}}{3})^{28}$$ $$A = 3^5 \times (7^{1/2})^{-10} \times \frac{(7^{1/2})^{28}}{3^{28}}$$ $$A = 3^5 \times 7^{-5} \times \frac{7^{14}}{3^{28}}$$ $$A = 3^{5-28} \times 7^{-5+14} = 3^{-23} \times 7^9$$

$$a = (-\sqrt{3})^5 \times 3^7 = -(3^{1/2})^5 \times 3^7$$ $$= -3^{5/2} \times 3^7 = -3^{5/2+7} = -3^{19/2}$$

$$b = 36 \times 15^{-2} = 6^2 \times (3 \times 5)^{-2}$$ $$= (2 \times 3)^2 \times 3^{-2} \times 5^{-2}$$ $$= 2^2 \times 3^2 \times 3^{-2} \times 5^{-2} = 2^2 \times 5^{-2} = (\frac{2}{5})^2$$

$$c = \sqrt[3]{\frac{8^{10} + 4^{10}}{2^{10} + 1}} = \sqrt[3]{\frac{(2^3)^{10} + (2^2)^{10}}{2^{10} + 1}}$$ $$= \sqrt[3]{\frac{2^{30} + 2^{20}}{2^{10} + 1}} = \sqrt[3]{\frac{2^{20}(2^{10} + 1)}{2^{10} + 1}}$$ $$= \sqrt[3]{2^{20}} = (2^{20})^{1/3} = 2^{20/3}$$

$$e = \sqrt{5}^2 - (-\sqrt{5})^4 \times (-1)^7$$ $$e = 5 - (5^2) \times (-1) = 5 - 25 \times (-1)$$ $$e = 5 + 25 = 30$$

$$d = \frac{7^{-3} \times (2 \times 3)^2 \times 15^{-2} \times 5^4}{(2 \times 5^6) \times (7^{-1} \times 2)^{3}}$$ $$d = \frac{7^{-3} \times 2^2 \times 3^2 \times (3 \times 5)^{-2} \times 5^4}{2 \times 5^6 \times 7^{-3} \times 2^3}$$ $$d = \frac{7^{-3} \times 2^2 \times 3^2 \times 3^{-2} \times 5^{-2} \times 5^4}{2^{1+3} \times 5^6 \times 7^{-3}}$$ $$d = \frac{7^{-3} \times 2^2 \times 3^0 \times 5^2}{2^4 \times 5^6 \times 7^{-3}}$$ $$d = 2^{2-4} \times 5^{2-6} = 2^{-2} \times 5^{-4} = \frac{1}{2^2 \times 5^4} = \frac{1}{4 \times 625} = \frac{1}{2500}$$

$$f = \sqrt{0,00005 \times \sqrt{0,02}} = \sqrt{5 \times 10^{-5} \times \sqrt{2 \times 10^{-2}}}$$ $$f = \sqrt{5 \times 10^{-5} \times \sqrt{2} \times \sqrt{10^{-2}}} = \sqrt{5 \times 10^{-5} \times \sqrt{2} \times 10^{-1}}$$ $$f = \sqrt{5 \sqrt{2} \times 10^{-6}} = \sqrt{5 \sqrt{2}} \times \sqrt{10^{-6}}$$ $$f = \sqrt{5\sqrt{2}} \times 10^{-3} = \frac{\sqrt{5\sqrt{2}}}{1000}$$