$\frac{6a^4{b}^2}{35c^3}\ast <!--\; \ast \; -->\frac{14b^2}{a^7{c}^5}\ast <!--\; \ast \; -->\frac{5a^3{c}^8}{18b^4}=\frac{1}{1}\ast <!--\; \ast \; -->\frac{2}{1}\ast <!--\; \ast \; -->\frac{1}{3}=\frac{2}{3}$

$\frac{33m^8}{34n^8}:\frac{88m^4}{51n^4}:\frac{21m^6}{16n^2}=\frac{33m^8}{34n^8}\ast <!--\; \ast \; -->\frac{51n^4}{88m^4}\ast <!--\; \ast \; -->\frac{16n^2}{21m^6}=\frac{1}{n^2}\ast <!--\; \ast \; -->\frac{3}{1}\ast <!--\; \ast \; -->\frac{1}{7m^2}=\frac{3}{7m^2{n}^2}$

$\frac{36x^6}{49y^5}:\frac{24x^9}{25y^4}\ast <!--\; \ast \; -->\frac{7x^2}{30y}=\frac{36x^6}{49y^5}\ast <!--\; \ast \; -->\frac{25y^4}{24x^9}\ast <!--\; \ast \; -->\frac{7x^2}{30y}=\frac{1}{7y}\ast <!--\; \ast \; -->\frac{5}{2x}\ast <!--\; \ast \; -->\frac{1}{2y}=\frac{5}{28x{y}^2}$

$(\frac{m^{5}n}{3p^3}{)}^3:\frac{m^{10}{n}^5}{54p^8}=(\frac{m^{5}n}{3p^3}{)}^3\ast <!--\; \ast \; -->\frac{54p^8}{m^{10}{n}^5}=\frac{m^{15}{n}^3}{27p^9}\ast <!--\; \ast \; -->\frac{54p^8}{m^{10}{n}^5}=\frac{m^5}{p}\ast <!--\; \ast \; -->\frac{2}{n^2}=\frac{2m^5}{n^{2}p}$

$(\frac{2a^5}{y^6}{)}^4:(\frac{4a^6}{y^8}{)}^3=\frac{16a^{20}}{y^{24}}:\frac{64a^{18}}{y^{24}}=\frac{16a^{20}}{y^{24}}\ast <!--\; \ast \; -->\frac{y^{24}}{64a^{18}}=\frac{a^2}{1}\ast <!--\; \ast \; -->\frac{1}{4}=\frac{a^2}{4}$

$(-<!--\; -\; -->\frac{27x^3}{16y^5}{)}^2\ast <!--\; \ast \; -->(\frac{8y^3}{9x^2}{)}^3=\frac{(3^3{)}^2{x}^6}{(2^4{)}^2{y}^{10}}\ast <!--\; \ast \; -->\frac{(2^3{)}^3{y}^9}{(3^2{)}^3{x}^6}=\frac{3^6{x}^6}{2^8{y}^{10}}\ast <!--\; \ast \; -->\frac{2^9{y}^9}{3^6{x}^6}=\frac{1}{y}\ast <!--\; \ast \; -->\frac{2}{1}=\frac{2}{y}$