7) \(\dfrac{6^{15}\cdot9^{10}}{3^{34}\cdot2^{13}}=\dfrac{2^{15}\cdot3^{15}\cdot\left(3^2\right)^{10}}{3^{34}\cdot2^{13}}=\dfrac{2^{15}\cdot3^{35}}{3^{34}\cdot2^{13}}=2^2\cdot3=12\)
8) \(\dfrac{9^2\cdot2^{11}}{16^2\cdot6^3}=\dfrac{\left(3^2\right)^2\cdot2^{11}}{\left(2^4\right)^2\cdot2^3\cdot3^3}=\dfrac{3^4\cdot2^{11}}{2^{11}\cdot3^3}=3\)
9) \(\dfrac{4^5\cdot9^4}{8^3\cdot27^3}=\dfrac{\left(2^2\right)^5\cdot\left(3^2\right)^4}{\left(2^3\right)^3\cdot\left(3^3\right)^3}=\dfrac{2^{10}\cdot3^8}{2^9\cdot3^9}=\dfrac{2}{3}\)
10) \(\dfrac{27^4\cdot4^3}{9^5\cdot8^2}=\dfrac{\left(3^3\right)^4\cdot\left(2^2\right)^3}{\left(3^2\right)^5\cdot\left(2^3\right)^2}=\dfrac{3^{12}\cdot2^6}{3^{10}\cdot2^6}=3^2=9\)
11) \(\dfrac{3^{29}\cdot4^{16}}{27^9\cdot8^{11}}=\dfrac{3^{29}\cdot\left(2^2\right)^{16}}{\left(3^3\right)^9\cdot\left(2^3\right)^{11}}=\dfrac{3^{29}\cdot2^{32}}{3^{27}\cdot2^{33}}=\dfrac{3^2}{2}=\dfrac{9}{2}\)
7) \(\dfrac{6^{15}.9^{10}}{3^{34}.2^{13}}=\dfrac{\left(2.3\right)^{15}.\left(3^2\right)^{10}}{3^{34}.2^{13}}=\dfrac{2^{15}.3^{15}.3^{20}}{3^{34}.2^{13}}\)
\(=\dfrac{2^{15}.3^{35}}{3^{34}.2^{13}}=2^2.3=12\)
8) \(\dfrac{9^2.2^{11}}{16^2.6^3}=\dfrac{\left(3^2\right)^2.2^{11}}{\left(2^4\right)^2.\left(2.3\right)^3}=\dfrac{3^4.2^{11}}{2^8.2^3.3^3}\)
\(=\dfrac{2^{11}.3^4}{2^{11}.3^3}=3\)
9) \(\dfrac{4^5.9^4}{8^3.27^3}=\dfrac{\left(2^2\right)^5.\left(3^2\right)^4}{\left(2^3\right)^3.\left(3^3\right)^3}=\dfrac{2^{10}.3^8}{2^9.3^9}=\dfrac{2}{3}\)
10) \(\dfrac{27^4.4^3}{9^5.8^2}=\dfrac{\left(3^3\right)^4.\left(2^2\right)^3}{\left(3^2\right)^5.\left(2^3\right)^2}=\dfrac{3^{12}.2^6}{3^{10}.2^6}=3^2=9\)
11) \(\dfrac{3^{29}.4^{16}}{27^9.8^{11}}=\dfrac{3^{29}.\left(2^2\right)^{16}}{\left(3^3\right)^9.\left(2^3\right)^{11}}=\dfrac{3^{29}.2^{32}}{3^{27}.2^{33}}\)
\(=\dfrac{3^2}{2}=\dfrac{9}{2}\)
12) \(\dfrac{4^{20}.3^{35}}{2^{37}.27^{12}}=\dfrac{\left(2^2\right)^{20}.3^{35}}{2^{37}.\left(3^3\right)^{12}}=\dfrac{2^{40}.3^{35}}{2^{37}.3^{36}}\)
\(=\dfrac{2^3}{3}=\dfrac{8}{3}\)
13) \(\dfrac{6^7.4^2}{9^2.12^5}=\dfrac{\left(2.3\right)^7.\left(2^2\right)^2}{\left(3^2\right)^2.\left(3.2^2\right)^5}=\dfrac{2^7.3^7.2^4}{3^4.3^5.2^{10}}\)
\(=\dfrac{2^{11}.3^7}{2^{10}.3^9}=\dfrac{2}{3^2}=\dfrac{2}{9}\)
14) \(\dfrac{15^2.9^3}{25^3.27^2}=\dfrac{\left(3.5\right)^2.\left(3^2\right)^3}{\left(5^2\right)^3.\left(3^3\right)^2}=\dfrac{3^2.5^2.3^6}{5^6.3^6}\)
\(=\dfrac{3^8.5^2}{5^6.3^6}=\dfrac{3^2}{5^4}=\dfrac{9}{625}\)
15) \(\dfrac{5^4.20^4}{25^5.4^5}=\dfrac{5^4.\left(5.2^2\right)^4}{\left(5^2\right)^5.\left(2^2\right)^5}=\dfrac{5^4.5^4.2^8}{5^{10}.2^{10}}\)
\(=\dfrac{5^8.2^8}{5^{10}.2^{10}}=\dfrac{1}{5^2.2^2}=\dfrac{1}{100}\)
12) \(\dfrac{4^{20}\cdot3^{35}}{2^{37}\cdot27^{12}}=\dfrac{\left(2^2\right)^{20}\cdot3^{35}}{2^{37}\cdot\left(3^3\right)^{12}}=\dfrac{2^{40}\cdot3^{35}}{2^{37}\cdot3^{36}}=\dfrac{2^3}{3}=\dfrac{8}{3}\)
13) \(\dfrac{6^7\cdot4^2}{9^2\cdot12^5}=\dfrac{2^7\cdot3^7\cdot\left(2^2\right)^2}{\left(3^2\right)^2\cdot4^5\cdot3^5}=\dfrac{2^7\cdot3^7\cdot2^4}{3^4\cdot\left(2^2\right)^5\cdot3^5}=\dfrac{2^{11}\cdot3^7}{3^9\cdot2^{10}}=\dfrac{2}{3^2}=\dfrac{2}{9}\)
14) \(\dfrac{15^2\cdot9^3}{25^3\cdot27^2}=\dfrac{5^2\cdot3^2\cdot\left(3^2\right)^3}{\left(5^2\right)^3\cdot\left(3^3\right)^2}=\dfrac{5^2\cdot3^2\cdot3^6}{5^6\cdot3^6}=\dfrac{3^2}{5^4}=\dfrac{9}{625}\)
15) \(\dfrac{5^4\cdot20^4}{25^5\cdot4^5}=\dfrac{5^4\cdot5^4\cdot4^4}{\left(5^2\right)^5\cdot\left(2^2\right)^5}=\dfrac{5^8\cdot\left(2^2\right)^4}{5^{10}\cdot2^{10}}=\dfrac{5^8\cdot2^8}{5^{10}\cdot2^{10}}=\dfrac{1}{5^2\cdot2^2}=\dfrac{1}{100}\)
