$\text{a)}\frac{5^7\cdot 6^5}{{30}^5}=\frac{5^{2+5}\cdot 6^5}{{\left(5\cdot 6\right)}^5}=\frac{5^2\cdot \cancel{5^5}\cdot \cancel{6^5}}{\cancel{5^5}\cdot \cancel{6^5}}=5^2=25$  

---

$\text{b)}\frac{{21}^8}{3^7\cdot 7^9}=\frac{{\left(3\cdot 7\right)}^8}{3^7\cdot 7^9}=\frac{3^8\cdot 7^8}{3^7\cdot 7^9}=\frac{\cancel{3^7}\cdot 3\cdot \cancel{7^8}}{\cancel{3^7}\cdot \cancel{7^8}\cdot 7}=\frac{3}{7}$  

---

$\text{c)}\frac{{22}^5\cdot 4}{2^8\cdot {11}^4}=\frac{{\left(2\cdot 11\right)}^5\cdot 2^2}{2^8\cdot {11}^4}=\frac{2^5\cdot {11}^5\cdot 2^2}{2^8\cdot {11}^4}=\frac{2^5\cdot 2^2\cdot {11}^5}{2^8\cdot {11}^4}=\frac{\cancel{2^7}\cdot \cancel{{11}^4}\cdot 11}{\cancel{2^7}\cdot 2\cdot \cancel{{11}^4}}=$  

$=\frac{11}{2}=5\frac{1}{2}$  

---

$\text{d)}\frac{3^{10}\cdot {13}^{11}}{{13}^3\cdot {39}^9}=\frac{3^{10}\cdot {13}^{11}}{{13}^3\cdot {\left(3\cdot 13\right)}^9}=\frac{3^{10}\cdot {13}^{11}}{{13}^3\cdot 3^9\cdot {13}^9}=\frac{3^{10}\cdot {13}^{11}}{{13}^3\cdot {13}^9\cdot 3^9}=\frac{3^{10}\cdot {13}^{11}}{{13}^{12}\cdot 3^9}=$  

$=\frac{\cancel{3^9}\cdot 3\cdot \cancel{{13}^{11}}}{\cancel{{13}^{11}}\cdot 13\cdot \cancel{3^9}}=\frac{3}{13}$  

---

$\text{e)}\frac{{\left({42}^3\right)}^5}{7^{15}\cdot 6^{13}}=\frac{{\left({\left(7\cdot 6\right)}^3\right)}^5}{7^{15}\cdot 6^{13}}=\frac{{\left(7^3\cdot 6^3\right)}^5}{7^{15}\cdot 6^{13}}=\frac{\cancel{7^{15}}\cdot 6^{15}}{\cancel{7^{15}}\cdot 6^{13}}=\frac{\cancel{6^{13}}\cdot 6^2}{\cancel{6^{13}}}=6^2=36$  

---

$\text{f)}\frac{{\left(2^2\right)}^8\cdot {\left(7^5\right)}^3}{{14}^{14}}=\frac{2^{16}\cdot 7^{15}}{{\left(2\cdot 7\right)}^{14}}=\frac{2^{16}\cdot 7^{15}}{2^{14}\cdot 7^{14}}=\frac{\cancel{2^{14}}\cdot 2^2\cdot \cancel{7^{14}}\cdot 7}{\cancel{2^{14}}\cdot \cancel{7^{14}}}=2^2\cdot 7=4\cdot 7=28$  

---

$\text{g)}\frac{{35}^{-9}}{5^{-7}\cdot 7^{-10}}=\frac{{\left(5\cdot 7\right)}^{-9}}{5^{-7}\cdot 7^{-10}}=\frac{5^{-9}\cdot 7^{-9}}{5^{-7}\cdot 7^{-10}}=\frac{\cancel{5^{-7}}\cdot 5^{-2}\cdot \cancel{7^{-9}}}{\cancel{5^{-7}}\cdot \cancel{7^{-9}}\cdot 7^{-1}}=\frac{5^{-2}}{7^{-1}}=\frac{{\left(\frac{1}{5}\right)}^2}{\frac{1}{7}}=$  

$=\frac{1}{25}\cdot 7=\frac{7}{25}$  

---

$\text{h)}\frac{6^{-8}\cdot {11}^{-9}}{{66}^{-10}\cdot 6^4}=\frac{6^{-8}\cdot {11}^{-9}}{{\left(6\cdot 11\right)}^{-10}\cdot 6^4}=\frac{6^{-8}\cdot {11}^{-9}}{6^{-10}\cdot {11}^{-10}\cdot 6^4}=\frac{6^{-8}\cdot {11}^{-9}}{6^{-10}\cdot 6^4\cdot {11}^{-10}}=$  

$=\frac{6^{-8}\cdot \cancel{{11}^{-9}}}{6^{-6}\cdot \cancel{{11}^{-9}}\cdot {11}^{-1}}=\frac{6^{-6}\cdot 6^{-2}}{6^{-6}\cdot {11}^{-1}}=\frac{\cancel{6^{-6}}\cdot 6^{-2}}{\cancel{6^{-6}}\cdot {11}^{-1}}=\frac{6^{-2}}{{11}^{-1}}=\frac{{\left(\frac{1}{6}\right)}^2}{\frac{1}{11}}=$  

$=\frac{1}{36}\cdot 11=\frac{11}{36}$  

---

$\text{i)}\frac{{\left(3^4\right)}^{-5}\cdot {\left({13}^{-6}\right)}^3}{{39}^{-19}}=\frac{3^{-20}\cdot {13}^{-18}}{{\left(3\cdot 13\right)}^{-19}}=\frac{3^{-20}\cdot {13}^{-18}}{3^{-19}\cdot {13}^{-19}}=\frac{\cancel{3^{-19}}\cdot 3^{-1}\cdot \cancel{{13}^{-18}}}{\cancel{3^{-19}}\cdot \cancel{{13}^{-18}}\cdot {13}^{-1}}=\frac{3^{-1}}{{13}^{-1}}=\frac{\frac{1}{3}}{\frac{1}{13}}=$  

$=\frac{1}{3}\cdot 13=\frac{13}{3}=4\frac{1}{3}$