m: \(M=\dfrac{17}{23}:\left(-9\right)-\dfrac{1}{23}:\left(-9\right)\)
\(=\dfrac{17}{23}\cdot\dfrac{-1}{9}-\dfrac{1}{23}\cdot\dfrac{-1}{9}\)
\(=\dfrac{-1}{9}\left(\dfrac{17}{23}-\dfrac{1}{23}\right)=\dfrac{16}{23}\cdot\dfrac{-1}{9}=-\dfrac{16}{207}\)
n: \(N=\dfrac{-23}{3}\cdot\dfrac{14}{4}+\dfrac{-25}{3}\cdot\dfrac{23}{4}+5\dfrac{3}{4}\)
\(=\dfrac{-23}{3}\cdot\dfrac{14}{4}+\dfrac{-23}{3}\cdot\dfrac{25}{4}+\dfrac{23}{4}\)
\(=\dfrac{-23}{4}\left(\dfrac{14}{3}+\dfrac{25}{3}-1\right)=-\dfrac{23}{4}\cdot12=-23\cdot3=-69\)
p: \(P=\dfrac{4^8+8^5}{4^6+8^4}\)
\(=\dfrac{2^{16}+2^{15}}{2^{12}+2^{12}}\)
\(=\dfrac{2^{15}\left(2+1\right)}{2^{13}}=2^2\cdot3=12\)
q: \(Q=\dfrac{10^{20}-2^{20}+6^{20}}{15^{20}-3^{20}+9^{20}}\)
\(=\dfrac{2^{20}\cdot5^{20}-2^{20}\cdot1+2^{20}\cdot3^{20}}{3^{20}\cdot5^{20}-3^{20}\cdot1+3^{20}\cdot3^{20}}\)
\(=\dfrac{2^{20}\left(5^{20}-1+3^{20}\right)}{3^{20}\left(5^{20}-1+3^{20}\right)}=\dfrac{2^{20}}{3^{20}}=\left(\dfrac{2}{3}\right)^{20}\)
r: \(R=\dfrac{7}{8}:\left(\dfrac{4}{9}-\dfrac{1}{18}\right)+\dfrac{7}{8}:\left(\dfrac{1}{36}-\dfrac{5}{12}\right)\)
\(=\dfrac{7}{8}:\dfrac{4\cdot2-1}{18}+\dfrac{7}{8}:\dfrac{1-15}{36}\)
\(=\dfrac{7}{8}\cdot\dfrac{18}{7}+\dfrac{7}{8}\cdot\dfrac{36}{-14}\)
\(=\dfrac{9}{4}-\dfrac{9}{4}=0\)
s: \(S=\dfrac{4^6\cdot9^5+6^9\cdot120}{-6^{11}-8^4\cdot3^{12}}\)
\(=\dfrac{2^{12}\cdot3^{10}+2^9\cdot3^9\cdot2^3\cdot3\cdot5}{-2^{11}\cdot3^{11}-2^{12}\cdot3^{12}}\)
\(=-\dfrac{2^{12}\cdot3^{10}\left(1+5\right)}{2^{11}\cdot3^{11}\left(1+2\cdot3\right)}=-\dfrac{2}{3}\cdot\dfrac{6}{7}=-\dfrac{12}{21}=-\dfrac{4}{7}\)
