26: \(\dfrac{15^{13}\cdot4^{12}}{6^{12}\cdot10^{13}}=\dfrac{5^{13}\cdot3^{13}\cdot2^{24}}{2^{12}\cdot2^{13}\cdot3^{12}\cdot5^{13}}=\dfrac{3^{13}}{3^{12}}\cdot\dfrac{2^{24}}{2^{25}}=\dfrac{3}{2}\)
29: \(\dfrac{6^{15}\cdot3^{10}}{4^7\cdot9^{13}}=\dfrac{2^{15}\cdot3^{15}\cdot3^{10}}{3^{26}\cdot2^{14}}=\dfrac{2^{15}}{2^{14}}\cdot\dfrac{3^{25}}{3^{26}}=\dfrac{2}{3}\)
32: \(\dfrac{7^6\cdot9^3}{21^5\cdot49}=\dfrac{7^6\cdot\left(3^2\right)^3}{3^5\cdot7^5\cdot7^2}=\dfrac{7^6}{7^7}\cdot\dfrac{3^6}{3^5}=\dfrac{3}{7}\)
35: \(\dfrac{9^2\cdot27^4}{3\cdot81^3}=\dfrac{\left(3^2\right)^2\cdot\left(3^3\right)^4}{3\cdot\left(3^4\right)^3}=\dfrac{3^4\cdot3^{12}}{3\cdot3^{12}}=3^3=27\)
38: \(\dfrac{4^{21}\cdot\left(-3\right)^{40}}{6^{41}}=\dfrac{2^{42}\cdot3^{40}}{2^{41}\cdot3^{41}}=\dfrac{2}{3}\)
41: \(\dfrac{81^{20}\cdot25^{55}}{125^{36}\cdot9^{40}}=\dfrac{\left(3^4\right)^{20}\cdot\left(5^2\right)^{55}}{\left(5^3\right)^{36}\cdot\left(3^2\right)^{40}}=\dfrac{3^{80}\cdot5^{110}}{5^{108}\cdot3^{80}}=5^2=25\)
44: \(\dfrac{2^5\cdot8^4\cdot4^3}{16^6}=\dfrac{2^5\cdot\left(2^3\right)^4\cdot\left(2^2\right)^3}{\left(2^4\right)^6}=\dfrac{2^5\cdot2^{12}\cdot2^6}{2^{24}}=\dfrac{2^{23}}{2^{24}}=\dfrac{1}{2}\)
47: \(\dfrac{27^2\cdot25^5\cdot2^5}{6\cdot15^4\cdot18}=\dfrac{\left(3^3\right)^2\cdot5^{10}\cdot2^5}{2\cdot5^4\cdot3^4\cdot3^2\cdot2}\)
\(=\dfrac{3^6\cdot5^{10}\cdot2^5}{2^2\cdot5^4\cdot3^6}=5^6\cdot2^3=125000\)
50: \(\dfrac{2^{2012}\cdot3^{2013}\cdot5^{2014}}{6^{2012}\cdot25^{1007}}=\dfrac{2^{2012}\cdot3^{2013}\cdot5^{2014}}{2^{2012}\cdot3^{2012}\cdot5^{2014}}=3\)
53: \(\dfrac{16^3\cdot3^{10}+120\cdot6^9}{4^6\cdot3^{12}+6^{11}}\)
\(=\dfrac{\left(2^4\right)^3\cdot3^{10}+2^3\cdot3\cdot5\cdot2^9\cdot3^9}{2^{12}\cdot3^{12}+2^{11}\cdot3^{11}}\)
\(=\dfrac{2^{12}\cdot3^{10}+2^{12}\cdot3^{10}\cdot5}{2^{11}\cdot3^{11}\left(2\cdot3+1\right)}=\dfrac{2^{12}\cdot3^{10}\cdot6}{2^{11}\cdot3^{11}\cdot7}\)
\(=\dfrac{2}{3}\cdot\dfrac{6}{7}=\dfrac{2\cdot2}{7}=\dfrac{4}{7}\)
\(8.\dfrac{16^3\cdot8^5}{4^{12}}\\ =\dfrac{\left(2^4\right)^3\cdot\left(2^3\right)^5}{\left(2^2\right)^{12}}\\ =\dfrac{2^{12}\cdot2^{15}}{2^{24}}\\ =\dfrac{2^{27}}{2^{24}}\\ =2^3=8\\ 11.\dfrac{8^3\cdot6^5}{2^{12}\cdot27^2}\\ =\dfrac{\left(2^3\right)^3\cdot2^5\cdot3^5}{2^{12}\cdot\left(3^3\right)^2}\\ =\dfrac{2^{14}\cdot3^5}{2^{12}\cdot3^6}\\ =\dfrac{2^2}{3}\\ =\dfrac{4}{3}\\ 14.\dfrac{2^{11}\cdot9^2}{3^5\cdot16^2}\\ =\dfrac{2^{11}\cdot\left(3^2\right)^2}{3^5\cdot\left(2^4\right)^2}\\ =\dfrac{2^{11}\cdot3^4}{2^8\cdot3^5}\\ =\dfrac{2^3}{3}\\ =\dfrac{8}{3}\\ 17.\dfrac{8^{17}\cdot15^{23}}{12^{25}\cdot25^{11}}\\ =\dfrac{\left(2^3\right)^{17}\cdot5^{23}\cdot3^{23}}{\left(2^2\right)^{25}\cdot3^{25}\cdot\left(5^2\right)^{11}}\\ =\dfrac{2^{51}\cdot5^{23}\cdot3^{23}}{2^{50}\cdot5^{22}\cdot3^{25}}\\ =\dfrac{2\cdot5}{3^2}\\ =\dfrac{10}{9}\)
