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.
\[ B = \left(\frac{7}{5}\right)^{29} \times \left(-\frac{7}{5}\right)^8 \]
\[
B = \left(\frac{7}{5}\right)^{29} \times \left(-\frac{7}{5}\right)^8
\]
\[
\left(-\frac{7}{5}\right)^8 = \left(-1\right)^8 \times \left(\frac{7}{5}\right)^8 = 1 \times \left(\frac{7}{5}\right)^8
\]
\[
B = \left(\frac{7}{5}\right)^{29} \times \left(\frac{7}{5}\right)^8
\]
\[
a^n \times a^p = a^{n+p} \implies B = \left(\frac{7}{5}\right)^{29 + 8}
\]
\[
B = \left(\frac{7}{5}\right)^{37}
\]
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.
\[
D = \frac{\left(-\frac{3}{2}\right)^{-2} \times \left(\frac{5}{3}\right)^3}{\left(\frac{1}{2}\right)^5}
\]
\[
(a/b)^{-n} = (b/a)^n \implies \left(-\frac{3}{2}\right)^{-2} = \left(-\frac{2}{3}\right)^2
\]
\[
\left(-\frac{2}{3}\right)^2 = \frac{4}{9}
\]
\[
\left(\frac{5}{3}\right)^3 = \frac{125}{27}
\]
\[
\left(\frac{1}{2}\right)^5 = \frac{1}{32}
\]
\[
D = \frac{\frac{4}{9} \times \frac{125}{27}}{\frac{1}{32}} = \frac{500}{243} \times 32
\]
\[
D = \frac{16000}{243}
\]