1. $\dfrac{8x^2}{9}\cdot \dfrac{9}{2}\Rightarrow \dfrac{\cancel{2}\cdot 4\cdot x^2}{\cancel{9}}\cdot \dfrac{\cancel{9}}{\cancel{2}}\Rightarrow 4x^2$
2. $\dfrac{8x}{3}\div \dfrac{4x}{7}\Rightarrow \dfrac{8x}{3}\cdot \dfrac{7}{4x} \Rightarrow \dfrac{2\cdot \cancel{4}\cdot \cancel{x}}{3}\cdot \dfrac{7}{\cancel{4}\cdot \cancel{x}}\Rightarrow \dfrac{14}{3}$
3. $\dfrac{5x^2}{4}\cdot \dfrac{6}{5}\Rightarrow \dfrac{\cancel{5}\cdot x^2}{2\cdot \cancel{2}}\cdot \dfrac{\cancel{2}\cdot 3}{\cancel{5}}\Rightarrow \dfrac{3x^2}{2}$
4. $\dfrac{10p}{5}\div \dfrac{8}{10}\Rightarrow \dfrac{10p}{5}\cdot \dfrac{10}{8}\Rightarrow \dfrac{\cancel{2}\cdot 5\cdot p}{\cancel{5}}\cdot \dfrac{\cancel{2} \cdot \cancel{5}}{2\cdot \cancel{2} \cdot \cancel{2}}\Rightarrow \dfrac{5p}{2}$
5. $\dfrac{\cancel{(m-6)}}{7\cancel{(7m-5)}}\cdot \dfrac{5m\cancel{(7m-5)}}{\cancel{m-6}}\Rightarrow \dfrac{5m}{7}$
6. $\dfrac{7(n-2)}{10(n+3)}\div \dfrac{n-2}{(n+3)}\Rightarrow \dfrac{7\cancel{(n-2)}}{10\cancel{(n+3)}}\cdot \dfrac{\cancel{(n+3)}}{\cancel{n-2}}\Rightarrow \dfrac{7}{10}$
7. $\dfrac{7r}{7r(r+10)}\div \dfrac{r-6}{(r-6)^2}\Rightarrow \dfrac{7r}{7r(r+10)}\cdot \dfrac{(r-6)^2}{r-6}\Rightarrow \dfrac{\cancel{7r}}{\cancel{7r}(r+10)}\cdot \dfrac{(r-6)\cancel{(r-6)}}{\cancel{r-6}}\Rightarrow \dfrac{r-6}{r+10}$
8. $\dfrac{6x(x+4)}{(x-3)}\cdot \dfrac{(x-3)(x-6)}{6x(x-6)}\Rightarrow \dfrac{\cancel{6x}(x+4)}{\cancel{(x-3)}}\cdot \dfrac{\cancel{(x-3)}\cancel{(x-6)}}{\cancel{6x}\cancel{(x-6)}}\Rightarrow x+4$
9. $\dfrac{x-10}{35x+21}\div \dfrac{7}{35x+21}\Rightarrow \dfrac{x-10}{7\cancel{(5x+3)}}\cdot \dfrac{\cancel{7}\cancel{(5x+3)}}{\cancel{7}}\Rightarrow \dfrac{x-10}{7}$
10. $\dfrac{v-1}{4}\cdot \dfrac{4}{v^2-11v+10}\Rightarrow \dfrac{\cancel{v-1}}{\cancel{4}}\Rightarrow \dfrac{\cancel{4}}{\cancel{(v-1)}(v-10)}\Rightarrow \dfrac{1}{v-10}$
11. $\dfrac{x^2-6x-7}{x+5}\cdot \dfrac{x+5}{x-7}\Rightarrow \dfrac{\cancel{(x-7)}(x+1)}{\cancel{(x+5)}}\cdot \dfrac{\cancel{(x+5)}}{\cancel{(x-7)}}\Rightarrow x+1$
12. $\dfrac{1}{a-6}\cdot \dfrac{8a+80}{8}\Rightarrow \dfrac{1}{a-6}\cdot \dfrac{\cancel{8}(a+10)}{\cancel{8}}\Rightarrow \dfrac{a+10}{a-6}$
13. $\dfrac{4m+36}{m+9}\cdot \dfrac{m-5}{5m^2}\Rightarrow \dfrac{4\cancel{(m+9)}}{\cancel{m+9}}\cdot \dfrac{m-5}{5m^2}\Rightarrow \dfrac{4(m-5)}{5m^2}$
14. $\dfrac{2r}{r+6}\div \dfrac{2r}{74+42}\Rightarrow \dfrac{\cancel{2r}}{\cancel{r+6}}\cdot \dfrac{7\cancel{(r+6)}}{\cancel{2r}}\Rightarrow 7$
15. $\dfrac{n-7}{6n-12}\cdot \dfrac{12-6n}{n^2-13n+42}\Rightarrow \dfrac{\cancel{(n-7)}}{\cancel{6}(n-2)}\cdot \dfrac{\cancel{6}(2-n)}{(n-6)\cancel{(n-7)}}\Rightarrow \dfrac{-1\cancel{(n-2)}}{\cancel{(n-2)}(n-6)}\Rightarrow \dfrac{-1}{n-6}$
16. $\dfrac{x^2+11x+24}{6x^3+18x^2}\cdot \dfrac{6x^3+6x^2}{x^2+5x-24}\Rightarrow \dfrac{\cancel{(x+3)}\cancel{(x+8)}}{\cancel{6x^2}\cancel{(x+3)}}\cdot \dfrac{\cancel{6x^2}(x+1)}{\cancel{(x+8)}(x-3)}\Rightarrow \dfrac{x+1}{x-3}$
17. $\dfrac{27a+36}{9a+63}\div \dfrac{6a+8}{2}\Rightarrow \dfrac{\cancel{9}\cancel{(3a+4)}}{\cancel{9}(a+7)}\cdot \dfrac{\cancel{2}}{\cancel{2}\cancel{(3a+4)}}\Rightarrow$
$\dfrac{1}{a+7}$
18. $\dfrac{k-7}{k^2-k-12}\cdot \dfrac{7k^2-28k}{8k^2-56k}\Rightarrow \dfrac{\cancel{k-7}}{\cancel{(k-4)}(k+3)}\cdot \dfrac{7\cdot \cancel{k}\cancel{(k-4)}}{8\cdot \cancel{k}\cancel{(k-7)}}\Rightarrow \dfrac{7}{8(k+3)}$
19. $\dfrac{x^2-12x+32}{x^2-6x-16}\cdot \dfrac{7x^2+14x}{7x^2+21x}\Rightarrow \dfrac{\cancel{(x-8)}(x-4)}{\cancel{(x-8)}\cancel{(x+2)}}\cdot \dfrac{\cancel{7x}\cancel{(x+2)}}{\cancel{7x}(x+3)}\Rightarrow \dfrac{x-4}{x+3}$
20. $\dfrac{9x^3+54x^2}{x^2+5x-14}\cdot \dfrac{x^2+5x-14}{10x^2}\Rightarrow \dfrac{9\cancel{x^2}(x+6)}{10\cancel{x^2}}\Rightarrow \dfrac{9(x+6)}{10}$
21. $(10m^2+100m)\cdot \dfrac{18m^3-36m^2}{20m^2-40m}\Rightarrow \cancel{10m}(m+10)\cdot \dfrac{\cancel{2}\cdot 9m^2\cancel{(m-2)}}{\cancel{2}\cdot \cancel{10m}\cancel{(m-2)}}\Rightarrow$
$9m^2(m+10)$
22. $\dfrac{n-7}{n^2-2n-35}\div \dfrac{9n+54}{10n+50}\Rightarrow \dfrac{\cancel{n-7}}{\cancel{(n-7)}\cancel{(n+5)}}\cdot \dfrac{10\cancel{(n+5)}}{9(n+6)}\Rightarrow \dfrac{10}{9(n+6)}$
23. $\\ \dfrac{x^2-1}{2x-4}\cdot \dfrac{x^2-4}{x^2-x-2}\div \dfrac{x^2+x-2}{3x-6}\Rightarrow$
$\dfrac{\cancel{(x-1)}\cancel{(x+1)}}{2\cancel{(x-2)}}\cdot \dfrac{\cancel{(x+2)}\cancel{(x-2)}}{\cancel{(x-2)}\cancel{(x+1)}}\cdot \dfrac{3\cancel{(x-2)}}{\cancel{(x+2)}\cancel{(x-1)}}\Rightarrow \dfrac{3}{2}$
24. $\dfrac{a^3+b^3}{a^2+3ab+2b^2}\cdot \dfrac{3a-6b}{3a^2-3ab+3b^2}\div \dfrac{a^2-4b^2}{a+2b}\Rightarrow$
$\dfrac{\cancel{(a+b)}\cancel{(a^2-ab+b^2)}}{(a+2b)\cancel{(a+b)}}\cdot \dfrac{\cancel{3}\cancel{(a-2b)}}{\cancel{3}\cancel{(a^2-ab+b^2)}}\cdot \dfrac{\cancel{a+2b}}{\cancel{(a-2b)}\cancel{(a+2b)}}\Rightarrow \dfrac{1}{a+2b}$