\[\left( -\frac{3}{2} \right)^{-3} = -\left( \frac{2}{3} \right)^3 = -\frac{8}{27} \\ \left( -\frac{4}{6} \right) = -\frac{2}{3} \\ \text{إذن} a = \frac{-\frac{8}{27}}{-\frac{2}{3}} = \frac{8}{27} \times \frac{3}{2} = \frac{24}{54} = \frac{4}{9}\]

\[1 + \frac{3}{4} - 1 = \frac{3}{4} \\ \frac{1}{2} - \frac{5}{6} + 1 = \frac{1}{2} + 1 - \frac{5}{6} = \frac{3}{2} - \frac{5}{6} = \frac{9 - 5}{6} = \frac{4}{6} = \frac{2}{3} \\ b = \left( \frac{\frac{3}{4}}{\frac{2}{3}} \right)^{-2} = \left( \frac{3}{4} \times \frac{3}{2} \right)^{-2} = \left( \frac{9}{8} \right)^{-2} = \frac{64}{81}\]

\[\sqrt{1.62} \approx 1.272 \\ \sqrt{0.18} \approx 0.424 \\ \sqrt{\frac{0.04}{10^{-4}}} = \sqrt{400} = 20 \\ \text{إذن} c \approx \frac{1.272}{0.424} \times 20 \approx 3 \times 20 = 60\]

\[1000^4 = 10^{12} \\ \text{البسط :} \frac{8}{10^{12} \times 10^{13}} = \frac{8}{10^{25}} \\ \text{المقام :} 0.0002 \times 10^7 = 2 \times 10^{-4} \times 10^7 = 2 \times 10^3 \\ \text{العبارة :} \frac{8}{10^{25}} \div 2 \times 10^3 = \frac{4}{10^{25} \times 10^3} = \frac{4}{10^{28}} \\ \sqrt{\cdots} = 2 \times 10^{-14}\]

\[\left( \frac{3}{2} \right)^{-3} = \left( \frac{2}{3} \right)^3 = \frac{8}{27} \\ \sqrt{\frac{81}{64}} = \frac{9}{8} \\ \text{إذن} e = \frac{8}{27} \times \frac{9}{8} = \frac{72}{216} = \frac{1}{3}\]

\[\left( -\frac{4}{3} \right)^2 = \frac{16}{9} \\ \frac{9}{5} = \frac{9}{5} \\ 5^{-1} = \frac{1}{5} \\ \text{إذن} f = \frac{16}{9} \times \frac{9}{5} + \frac{1}{5} = \frac{16}{5} + \frac{1}{5} = \frac{17}{5}\]

\[(-7)^{-2} = \frac{1}{49} \\ (-7)^8 = 5764801 \\ (-7)^3 = -343 \\ \text{إذن} \frac{1}{49} \times 5764801 \div (-343) = \frac{5764801}{49 \times -343}\]

\[\left( \frac{-5}{3} \right)^{-3} = \left( \frac{-3}{5} \right)^3 = \frac{-27}{125} \\ \left( \frac{-27}{125} \right)^5 = \frac{-27^5}{125^5} \\ \text{ثمّ} \times \frac{25}{9}\]

\[(-\frac{3}{4})^{-5} = (-\frac{4}{3})^5 = \frac{-1024}{243} \\ \text{العامل الآخر :} (-\frac{3}{4})^{17} = \frac{(-3)^{17}}{4^{17}} \\ \text{الجداء : نبسّط القوى}\]

\[\sqrt{\frac{16}{25}} = \frac{4}{5} \\ (\frac{4}{5})^4 = \frac{256}{625} \\ (\frac{5}{4})^{-7} = (\frac{4}{5})^7 \\ \text{إذن} \frac{256}{625} \times \frac{16384}{78125}\]

\[\sqrt{30000} \approx 173.2 \\ \frac{0.03}{173.2} \approx 0.0001732\]

\[(7^3)^{12} = 7^{36} \\ 7^{-30} = 7^{-30} \\ \text{إذن} 7^{36} \times 7^{-30} = 7^{6}\]

\[\left( 2^{-1} \right)^{-3} = 2^3 = 8\]

\[2^{-3} \times 2^{-5} \times 2^8 = 2^{-8 + 8} = 2^0 = 1\]

\[\left( \frac{1}{2} \right)^{-3} = 2^3 = 8 \\ \left( \frac{1}{8} \right)^{-1} = 8 \\ \text{إذن} 8 \times 2^4 \div 8 = 2^4 = 16\]

\[\left( \frac{-1}{3} \right)^3 = \frac{-1}{27} \\ \left( \frac{-1}{3} \right)^2 = \frac{1}{9} \\ \div \left( \frac{-1}{3} \right) = \times \left( \frac{-3}{1} \right) \\ \text{جداء الثلاثة :} \frac{-1}{27} \times \frac{1}{9} \times \left( -3 \right) = \frac{1}{81} \times (-3) = -\frac{1}{27}\]

\[10^3 \times 10^{-7} = 10^{-4} \\ 10^{-4} \times 1000 = 10^{-4} \times 10^3 = 10^{-1} = 0.1\]

\[9^2 = 81 \\ \sqrt{81 + 19} = \sqrt{100} = 10 \\ \sqrt{\frac{25}{4}} = \frac{5}{2} \\ \text{إذن} f = 10 + \frac{5}{2} = \frac{25}{2} = 12.5\]

\[2^{-16} \times 5^{-16} = (2 \times 5)^{-16} = 10^{-16}\]

\[\left( \frac{4}{7} \right)^3 \times \left( \frac{4}{7} \right)^{-12} = \left( \frac{4}{7} \right)^{-9} = \left( \frac{7}{4} \right)^9\]

\[10^{-4} = \frac{1}{10^4} \\ (10^{-4})^{-2} = 10^8 \\ C = 100 \times 10^8 \div 10^{-12} = 100 \times 10^{20} = 10^{22}\]

\[(\frac{2}{5})^3 \div (\frac{2}{5})^7 = (\frac{2}{5})^{-4} = (\frac{5}{2})^4 = \frac{625}{16}\]

\[\left( \frac{5}{3} \right)^{-5} = \left( \frac{3}{5} \right)^5 \\ \left( \cdots \right)^{-3} = \left( \frac{5}{3} \right)^{15} \\ \text{إذن} \left( \frac{5}{3} \right)^{15} \times \frac{25}{9}\]

\[(\frac{2}{5})^{-3} = (\frac{5}{2})^3 \\ \left( \cdots \right)^5 = (\frac{5}{2})^{15} \\ \text{ثمّ} \times (\frac{2}{5})^9 = (\frac{5}{2})^{15} \times (\frac{2}{5})^9 = (\frac{5}{2})^{6} \\ \times \frac{4}{25} = \frac{4}{25} \times (\frac{5}{2})^6\]