31: \(\dfrac{6^4\cdot9^2}{6^5\cdot8^3}=\dfrac{1}{6}\cdot\dfrac{3^4}{2^6}=\dfrac{3^3}{2^7}=\dfrac{27}{128}\)
32: \(\dfrac{7^5\cdot9^3}{21^5\cdot49}=\dfrac{7^5\cdot\left(3^2\right)^3}{7^5\cdot3^5\cdot7^2}=\dfrac{1}{7^2}\cdot\dfrac{3^6}{3^5}=\dfrac{3}{49}\)
33 \(\dfrac{20^5\cdot5^{10}}{100^4}=\dfrac{\left(5\cdot2^2\right)^5\cdot5^{10}}{\left(5^2\cdot2^2\right)^4}=\dfrac{5^{15}\cdot2^{10}}{2^8\cdot5^8}=5^7\cdot2^2\)
34: \(\dfrac{\left(-5\right)^2\cdot20^4}{8^2\cdot125}=\dfrac{5^2\cdot5^2\cdot2^8}{\left(2^3\right)^2\cdot5^3}=\dfrac{5^4}{5^3}\cdot\dfrac{2^8}{2^6}=5\cdot2^2=20\)
35: \(\dfrac{9^2\cdot27^4}{3\cdot81^3}=\dfrac{\left(3^2\right)^2\cdot\left(3^3\right)^4}{3\cdot\left(3^4\right)^3}=\dfrac{3^4\cdot3^{12}}{3\cdot3^{12}}=\dfrac{3^{16}}{3^{13}}=3^3=27\)
36: \(\dfrac{9^{16}\cdot5^{29}}{125^9\cdot27^{11}}=\dfrac{\left(3^2\right)^{16}\cdot5^{29}}{\left(3^3\right)^{11}\cdot\left(5^3\right)^9}\)
\(=\dfrac{3^{32}}{3^{33}}\cdot\dfrac{5^{29}}{5^{27}}\)
\(=\dfrac{5^2}{3}=\dfrac{25}{3}\)
