M=\(\frac{9^4\times27^5\times3^6\times3^4}{3^8\times81^4\times243\times8^2}\)
N=4\(\frac{4^6\times9^5\times6^9\times120}{8^4\times3^{12}-6^{11}}\)
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a) \(\frac{9}{33-3}=\frac{1}{3}\)
b) \(\frac{7}{100+6\times100}=\frac{1}{100}\)
c) \(\frac{11\times22+33\times36+55\times60}{22\times24+66\times72+110\times120}=\frac{1}{4}\)
d) \(\frac{9^4\times27^5\times3^6\times4^4}{3^8\times81^4\times243\times8^2}=4\)
e) \(\frac{199919991999}{200020002000}=\frac{1999}{2000}\)
\(=\frac{2^{12}\times3^{10}+2^9\times3^9\times2^3\times3\times5}{-2^{12}\times3^{12}-2^{11}\times3^{11}}\)
\(=\frac{2^{12}\times3^{10}+2^{12}\times3^{10}\times5}{-2^{11}\times3^{11}\times\left(2\times3+1\right)}\)
\(=\frac{2^{12}\times3^{10}\times\left(1+5\right)}{-2^{11}\times3^{11}\times7}\)
\(=\frac{2^{13}\times3^{11}}{-2^{11}\times3^{11}\times7}\)
\(=\frac{-4}{7}\)
\(\dfrac{5\times4^{15}\times9^9-4\times3^{20}\times8^9}{5\times2^{10}\times6^{19}-7\times2^{29}\times27^6}\\ =\dfrac{5\times2^{30}\times3^{18}-2^2\times3^{20}\times2^{27}}{5\times2^{10}\times3^{19}\times2^{19}-7\times2^{29}\times3^{18}}\\ =\dfrac{5\times2^{30}\times3^{18}-2^{29}\times3^{20}}{5\times2^{29}\times3^{19}-7\times2^{29}\times3^{18}}\\ =\dfrac{2^{29}\times3^{18}\times\left(5\times2-3^2\right)}{2^{29}\times3^{18}\times\left(5\times3-7\right)}\\ =\dfrac{10-9}{15-7}\\ =\dfrac{1}{8}\)