1.
\[ C = \frac{\left(\frac{7}{3}\right)^{24}}{\left(\frac{7}{3}\right)^{14}} \]
\[ C = \frac{\left(\frac{7}{3}\right)^{24}}{\left(\frac{7}{3}\right)^{14}} \]
\[ \frac{a^n}{a^p} = a^{n-p} \implies C = \left(\frac{7}{3}\right)^{24-14} \]
\[ C = \left(\frac{7}{3}\right)^{10} \]
2.
\[ B = \left(\frac{7}{5}\right)^{29} \times \left(-\frac{7}{5}\right)^8 \]
\[ B = \left(\frac{7}{5}\right)^{29} \times \left(-\frac{7}{5}\right)^8 \]
\[ \left(-\frac{7}{5}\right)^8 = (-1)^8 \times \left(\frac{7}{5}\right)^8 = 1 \times \left(\frac{7}{5}\right)^8 \]
\[ B = \left(\frac{7}{5}\right)^{29} \times \left(\frac{7}{5}\right)^8 \]
\[ a^n \times a^p = a^{n+p} \implies B = \left(\frac{7}{5}\right)^{29+8} \]
\[ B = \left(\frac{7}{5}\right)^{37} \]
3.
\[ A = \left(\frac{9}{5}\right)^{-11} \times \left(\frac{9}{5}\right)^{20} \]
\[ A = \left(\frac{9}{5}\right)^{-11} \times \left(\frac{9}{5}\right)^{20} \]
\[ a^n \times a^p = a^{n+p} \implies A = \left(\frac{9}{5}\right)^{-11+20} \]
\[ A = \left(\frac{9}{5}\right)^9 \]
4.
\[ F = \frac{0.001^4 \times 1000^2}{100^3 \times 0.01^5} \]
\[ 0.001 = 10^{-3} \]
\[ (a^n)^p = a^{n \times p} \implies 0.001^4 = (10^{-3})^4 = 10^{-3 \times 4} = 10^{-12} \]
\[ 1000 = 10^3 \]
\[ (10^3)^2 = 10^{3 \times 2} = 10^6 \]
\[ 100 = 10^2 \]
\[ (10^2)^3 = 10^{2 \times 3} = 10^6 \]
\[ 0.01 = 10^{-2} \]
\[ (10^{-2})^5 = 10^{-2 \times 5} = 10^{-10} \]
\[ F = \frac{10^{-12} \times 10^6}{10^6 \times 10^{-10}} \]
\[ a^n \times a^p = a^{n+p} \implies \text{البسط} = 10^{-12+6} = 10^{-6} \]
\[ \text{المقام} = 10^{6 + (-10)} = 10^{-4} \]
\[ \frac{a^n}{a^p} = a^{n-p} \implies F = 10^{-6 - (-4)} = 10^{-2} \]
5.
\[ E = \left(\frac{27}{8}\right)^{-2} \times \left(\frac{3}{2}\right)^4 \]
\[ 27 = 3^3, \quad 8 = 2^3 \]
\[ \frac{27}{8} = \frac{3^3}{2^3} \]
\[ (a^n / b^n)^p = (a/b)^{n \times p} \implies \left(\frac{3^3}{2^3}\right)^{-2} = \left(\frac{3}{2}\right)^{3 \times (-2)} = \left(\frac{3}{2}\right)^{-6} \]
\[ (a^n)^p = a^{n \times p} \implies \left(\frac{3}{2}\right)^{-6} = \left(\frac{2}{3}\right)^6 \]
\[ E = \left(\frac{2}{3}\right)^6 \times \left(\frac{3}{2}\right)^4 \]
\[ a^n \times a^p = a^{n+p} \implies E = 2^{6} \times 3^{-6} \times 3^{4} \times 2^{-4} \]
\[ E = 2^{6-4} \times 3^{-6+4} = 2^{2} \times 3^{-2} \]
\[ E = \frac{4}{9} \]
6.
\[ D = \frac{\left(-\frac{3}{2}\right)^{-2} \times \left(\frac{5}{3}\right)^3}{\left(\frac{1}{2}\right)^5} \]
\[ (a/b)^{-n} = (b/a)^n \implies \left(-\frac{3}{2}\right)^{-2} = \left(-\frac{2}{3}\right)^2 \]
\[ \left(-\frac{2}{3}\right)^2 = \frac{4}{9} \]
\[ \left(\frac{5}{3}\right)^3 = \frac{125}{27} \]
\[ \left(\frac{1}{2}\right)^5 = \frac{1}{32} \]
\[ D = \frac{\frac{4}{9} \times \frac{125}{27}}{\frac{1}{32}} = \frac{500}{243} \times 32 \]
\[ D = \frac{16000}{243} \]