\(31,\dfrac{6^4.9^2}{6^5.8^3}=\dfrac{1.\left(3^2\right)^2}{6.\left(2^3\right)^3}=\dfrac{3^4}{6.2^9}=\dfrac{3^4}{2.3.2^9}=\dfrac{3^3}{2.2^9}=\dfrac{27}{1024}\)
\(32,\dfrac{7^6.9^3}{21^5.49}=\dfrac{7^6.\left(3^2\right)^3}{\left(3.7\right)^5.7^2}=\dfrac{7^6.3^6}{3^5.7^5.7^2}=\dfrac{7^6.3^6}{3^5.7^7}=\dfrac{1.3}{1.7}=\dfrac{3}{7}\)
\(33,\dfrac{20^5.5^{10}}{100^4}=\dfrac{20^5.5^5.5^5}{100^4}=\dfrac{\left(20.5\right)^5.5^5}{100^4}=\dfrac{100^5.5^5}{100^4}=100.5^5=312500\)
\(34,\dfrac{\left(-5\right)^2.20^4}{8^2.125}=\dfrac{5^2.\left(5.4\right)^4}{\left(2^3\right)^2.5^3}=\dfrac{5^2.5^4.4^4}{2^6.5^3}=\dfrac{5^6.\left(2^2\right)^4}{2^6.5^3}=\dfrac{5^3.2^8}{2^6}=5^3.2^2=125.4=500\)
\(35,\dfrac{9^2.27^4}{3.81^3}=\dfrac{\left(3^2\right)^2.\left(3^3\right)^4}{3.\left(3^4\right)^3}=\dfrac{3^4.3^{12}}{3.3^{12}}=\dfrac{3^3.1}{1.1}=3^3=27\)
\(36,\dfrac{9^{16}.5^{29}}{125^9.27^{11}}=\dfrac{\left(3^2\right)^{16}.5^{29}}{\left(5^3\right)^9.\left(3^3\right)^{11}}=\dfrac{3^{32}.5^{29}}{5^{27}.3^{33}}=\dfrac{1.5^2}{1.3}=\dfrac{5^2}{3}=\dfrac{25}{3}\)
