1) \(c = \sqrt{0.18 \times 10^{-4}}\)
\[0.18 = \frac{18}{100} = \frac{9 \cdot 2}{10^2} \\ \Rightarrow c = \sqrt{0.18 \times 10^{-4}} = \sqrt{18 \times 10^{-6}} = \sqrt{9 \cdot 2} \cdot 10^{-3} = 3 \sqrt{2} \cdot 10^{-3}\]
2) \(b = \left(\frac{1}{2} + \frac{3}{4} - \frac{1}{5}\right)^3\)
\[\text{نحتفظ بالصيغة الكسريّة :} \\ \text{نوحّد المقامات ثمّ :} \left(\frac{1}{2} + \frac{3}{4} - \frac{1}{5}\right)^3 = \left( \text{كسر} \right)^3 = \text{قوّة كسر}\]
4) \(f = \left(-\frac{4}{3}\right)^2 \times 6 + 5^{-1}\)
\[\left(-\frac{4}{3}\right)^2 = \left(\frac{4}{3}\right)^2 = \frac{4^2}{3^2} = \frac{16}{9} \\ 5^{-1} = \frac{1}{5} \\ \text{النتيجة النهائيّة في صيغة :} \frac{16}{9} \cdot 6 + \frac{1}{5}\]
7) \(c = \frac{(-7)^2 \times (-7)^8}{(-7)^7}\)
\[(-7)^2 \cdot (-7)^8 = (-7)^{2+8} = (-7)^{10} \\ \Rightarrow \frac{(-7)^{10}}{(-7)^7} = (-7)^{10 - 7} = (-7)^3\]
9) \(a = \left(\frac{3}{4}\right)^5 \times \left(\frac{3}{4}\right)^7\)
\[\left(\frac{3}{4}\right)^5 \cdot \left(\frac{3}{4}\right)^7 = \left(\frac{3}{4}\right)^{5 + 7} = \left(\frac{3}{4}\right)^{12}\]
11) \(b = 2^3 \times 2^{-5} \times 2^2\)
\[2^3 \cdot 2^{-5} \cdot 2^2 = 2^{3 - 5 + 2} = 2^0 = 1\]
14) \(e = 10^9 \times 10^{-7} \times 1000\)
\[1000 = 10^3 \\ \Rightarrow 10^9 \cdot 10^{-7} \cdot 10^3 = 10^{9 - 7 + 3} = 10^5\]
16) \(c = \frac{100 \times (10^{-1})^2}{10^{12}}\)
\[(10^{-1})^2 = 10^{-2} \\ 100 = 10^2 \\ \Rightarrow \frac{10^2 \cdot 10^{-2}}{10^{12}} = \frac{10^0}{10^{12}} = 10^{-12}\]
18) \(A = 2^{-16} \times 5^{-16}\)
\[2^{-16} \cdot 5^{-16} = (2 \cdot 5)^{-16} = 10^{-16}\]
19) \(F = \left[\left(\frac{2}{5}\right)^3\right] \times \left(\frac{2}{5}\right)^6 \times \frac{4}{25}\)
\[\left(\frac{2}{5}\right)^3 \cdot \left(\frac{2}{5}\right)^6 = \left(\frac{2}{5}\right)^9 \\ \frac{4}{25} = \left(\frac{2}{5}\right)^2 \\ \Rightarrow \left(\frac{2}{5}\right)^9 \cdot \left(\frac{2}{5}\right)^2 = \left(\frac{2}{5}\right)^{11}\]