\(1,\dfrac{4^2.4^3}{2^{10}}=\dfrac{\left(2^2\right)^2.\left(2^2\right)^3}{2^{10}}=\dfrac{2^4.2^6}{2^{10}}=\dfrac{2^{10}}{2^{10}}=1\)
\(2,\dfrac{2^7.9^3}{6^5.8^2}=\dfrac{2^7.\left(3^2\right)^3}{\left(2.3\right)^5.\left(2^3\right)^2}=\dfrac{2^7.3^6}{2^5.3^5.2^6}=\dfrac{2^7.3^6}{2^{11}.3^5}=\dfrac{3}{2^4}=\dfrac{3}{16}\)
\(3.\dfrac{10^5.7^3}{14^2.20^4}=\dfrac{\left(2.5\right)^5.7^3}{\left(2.7\right)^2.\left(2^2.5\right)^4}=\dfrac{2^5.5^5.7^3}{2^2.7^2.2^8.5^4}=\dfrac{2^5.5^5.7^3}{2^{10}.7^2.5^4}=\dfrac{5.7}{2^5}=\dfrac{35}{32}\)
\(4,\dfrac{8^3.6^5}{2^{12}.27^2}=\dfrac{\left(2^3\right)^3.\left(2.3\right)^5}{2^{12}.\left(3^3\right)^2}=\dfrac{2^9.2^5.3^5}{2^{12}.3^6}=\dfrac{2^{14}.3^5}{2^{12}.3^6}=\dfrac{2^2}{3}=\dfrac{4}{3}\)
\(5,\dfrac{5^4.20^4}{25^5.4^5}=\dfrac{5^4.\left(4.5\right)^4}{\left(5^2\right)^5.4^5}=\dfrac{5^4.4^4.5^4}{5^{10}.4^5}=\dfrac{5^8.4^4}{5^{10}.4^5}=\dfrac{1}{5^2.4}=\dfrac{1}{25.4}=\dfrac{1}{100}\)
\(6,\dfrac{3^7.9^3}{81.27^2}=\dfrac{3^7.\left(3^2\right)^3}{3^4.\left(3^3\right)^2}=\dfrac{3^7.3^6}{3^4.3^6}=3^3=27\)
\(7,\dfrac{45^{10}.5^{25}}{75^{13}}=\dfrac{\left(9.5\right)^{10}.5^{25}}{\left(15.5\right)^{13}}=\dfrac{9^{10}.5^{10}.5^{25}}{15^{13}.5^{13}}=\dfrac{\left(3^2\right)^{10}.5^{35}}{\left(3.5\right)^{13}.5^{13}}=\dfrac{3^{20}.5^{35}}{3^{13}.5^{13}.5^{13}}=\dfrac{3^{20}.5^{35}}{3^{13}.5^{26}}=3^7.5^9=2187.1953125=4271484375\)
\(8,\dfrac{4^{21}.\left(-3\right)^{40}}{6^{41}}=\dfrac{\left(2^2\right)^{21}.3^{40}}{\left(2.3\right)^{41}}=\dfrac{2^{42}.3^{40}}{2^{41}.3^{41}}=\dfrac{2}{3}\)
\(9,\dfrac{\left(-5\right)^2.\left(-5\right)^3.16}{5^4.\left(-2\right)^4}=\dfrac{\left(-5\right)^5.2^4}{5^4.2^4}=-5\)
