تمارين القوى مع التصحيحات المفصلة
\[a = \sqrt{\frac{81}{16}}\]
\[\sqrt{\frac{81}{16}} = \frac{9}{4}\]
\[b = (-2)^8 \times (2017)^0 \times (-5)^8\]
\[(-2)^8 = 256 \\ (-5)^8 = 390625 \\ (2017)^0 = 1 \\ b = 256 \times 1 \times 390625 = 100000000\]
\[c = \sqrt{\frac{75}{27}} \div 3^{-2}\]
\[\sqrt{\frac{75}{27}} = \sqrt{\frac{25 \cdot 3}{3 \cdot 9}} = \sqrt{\frac{25}{9}} = \frac{5}{3} \\ 3^{-2} = \frac{1}{9} \\ c = \frac{5}{3} \div \frac{1}{9} = \frac{5}{3} \times 9 = 15\]
\[d = \frac{2}{3} \left( \frac{2}{3} \right)^3\]
\[d = \frac{2}{3} \times \left( \frac{2}{3} \right)^3 = \left( \frac{2}{3} \right)^4 = \frac{16}{81}\]
\[e = \left( \frac{1}{3} \right)^{-2} \times \left( \frac{1}{2} \right)^{-1} \div \left( \frac{1}{2} \right)^{-1} \times \frac{5}{2}\]
\[\left( \frac{1}{3} \right)^{-2} = 3^2 = 9 \\ \left( \frac{1}{2} \right)^{-1} = 2 \\ \div \left( \frac{1}{2} \right)^{-1} = \div 2 \\ \Rightarrow e = 9 \times 2 \div 2 \times \frac{5}{2} = 9 \times \frac{5}{2} = \frac{45}{2}\]
\[f = \left( \frac{3^7}{7^3} \right)^0 + \frac{5}{2} + \frac{\sqrt{16}}{\sqrt{4}}\]
\[\left( \frac{3^7}{7^3} \right)^0 = 1 \\ \sqrt{16} = 4 , \sqrt{4} = 2 \\ \frac{4}{2} = 2 \\ f = 1 + \frac{5}{2} + 2 = \frac{9}{2} + 2 = \frac{13}{2}\]
\[(-5)^9 \times (2017)^0 \times (-2)^9\]
\[(-5)^9 = -1953125 \\ (-2)^9 = -512 \\ (2017)^0 = 1 \\ \text{النتيجة} = (-1953125) \times 1 \times (-512) = 1000000000\]
\[10^2 \times 10^{-5} \times 1000\]
\[10^2 = 100 \\ 10^{-5} = 1/100000 \\ 1000 = 10^3 \\ \text{النتيجة} = 100 \times 1/100000 \times 1000 = \frac{100000}{100000} = 1\]
\[\left( \frac{5}{2} \right)^2\]
\[\left( \frac{5}{2} \right)^2 = \frac{25}{4}\]
\[\left[ (-2)^{-2} \right]^{-2}\]
\[(-2)^{-2} = \frac{1}{4} \\ \left( \frac{1}{4} \right)^{-2} = 4^2 = 16\]
\[\sqrt{16}\]
\[\sqrt{16} = 4\]
\[\sqrt{\frac{81}{49}}\]
\[\sqrt{\frac{81}{49}} = \frac{9}{7}\]
\[A = 3^5 \times 31^3 \times 3^2\]
\[A = 3^5 \times 3^2 \times 31^3 = 3^{5+2} \times 31^3 = 3^7 \times 31^3\]
\[B = (-7)^{-20} \times (10)^{-20}\]
\[B = (-7)^{-20} \times 10^{-20} = \left( -7 \times 10 \right)^{-20} = (-70)^{-20}\]
\[C = \frac{100 \times [10^{-11}]^{-2}}{10^{-15}}\]
\[10^{-11} = \frac{1}{10^{11}} \\ [10^{-11}]^{-2} = 10^{22} \\ 100 \times 10^{22} = 10^2 \times 10^{22} = 10^{24} \\ \div 10^{-15} = 10^{24} \times 10^{15} = 10^{39}\]
\[D = \left[ \left( \frac{-2}{3} \right)^2 \right]^7 \times \left( \frac{-8}{27} \right)\]
\[\left( \left( \frac{-2}{3} \right)^2 \right)^7 = \left( \frac{4}{9} \right)^7 \\ \left( \frac{4}{9} \right)^7 \times \left( \frac{-8}{27} \right) = \frac{4^7 \cdot -8}{9^7 \cdot 27} = \frac{-8 \cdot 4^7}{9^7 \cdot 27}\]
\[a = \left( \frac{5}{2} \right)^3 \times \left( \frac{4}{25} \right)^{-2}\]
\[a = \left( \frac{5}{2} \right)^3 \times \left( \frac{4}{25} \right)^{-2} = \left( \frac{5}{2} \right)^3 \times \left( \frac{25}{4} \right)^2 = \frac{125}{8} \times \frac{625}{16} = \frac{78125}{128}\]
\[b = \left( \frac{1}{5} \right)^3 \times 27^{-4}\]
\[\left( \frac{1}{5} \right)^3 = \frac{1}{125} \\ 27^{-4} = \frac{1}{531441} \\ b = \frac{1}{125 \cdot 531441}\]
\[c = \frac{64 \times 4}{4^{-3}}\]
\[c = \frac{64 \cdot 4}{4^{-3}} = \frac{256}{4^{-3}} = 256 \cdot 4^3 = 256 \cdot 64 = 16384\]
\[d = \left( \frac{8}{27} \right)^3 \times \left( \frac{3}{2} \right)^6\]
\[\left( \frac{8}{27} \right)^3 = \frac{512}{19683} \\ \left( \frac{3}{2} \right)^6 = \frac{729}{64} \\ d = \frac{512 \cdot 729}{19683 \cdot 64}\]
\[e = \frac{9^5 \times 81 \times 4^{-3}}{(-2)^{-2} \times 5^{-2}}\]
\[9^5 = 59049 , 81 = 3^4 , 4^{-3} = 1/64 \\ (-2)^{-2} = 1/4 , 5^{-2} = 1/25 \\ e = \frac{59049 \cdot 81 \cdot 1/64}{1/4 \cdot 1/25} = \frac{59049 \cdot 81}{64} \cdot 100 = \cdots\]
\[f = \left( \frac{-1}{5} \right)^{-2} \times 125^3\]
\[\left( \frac{-1}{5} \right)^{-2} = \left( \frac{5}{1} \right)^2 = 25 \\ 125^3 = 1953125 \\ f = 25 \times 1953125\]
\[g = 2 \times 7^5 + 5 \times 7^{-5}\]
\[g = 2 \cdot 7^5 + 5 \cdot 7^{-5} = 2 \cdot 16807 + \frac{5}{16807} = 33614 + \frac{5}{16807}\]
\[h = \frac{64^2 \times 2^4}{(2^{-2})^5 \times 2^{-2}}\]
\[64^2 = 4096 \\ (2^{-2})^5 = 2^{-10} \\ \times 2^{-2} = 2^{-12} \\ \text{إذن} h = \frac{4096 \cdot 2^4}{2^{-12}} = 4096 \cdot 16 \cdot 2^{12} = 65536 \cdot 4096\]