a: \(\dfrac{3}{5}:\left(-\dfrac{1}{15}-\dfrac{1}{6}\right)+\dfrac{3}{5}:\left(-\dfrac{1}{3}-1\dfrac{1}{15}\right)\)
\(=\dfrac{3}{5}:\left(-\dfrac{2}{30}-\dfrac{5}{30}\right)+\dfrac{3}{5}:\left(-\dfrac{5}{15}-\dfrac{16}{15}\right)\)
\(=\dfrac{3}{5}\cdot\dfrac{-30}{7}+\dfrac{3}{5}\cdot\dfrac{-15}{21}\)
\(=\dfrac{3}{5}\cdot\left(-\dfrac{30}{7}+\dfrac{-5}{7}\right)=\dfrac{3}{5}\cdot\left(-5\right)=-3\)
b: \(\sqrt{\dfrac{4}{25}}\cdot\left(-\dfrac{1}{25}\right)^0+\left|\dfrac{9}{10}\right|\cdot0,\left(6\right)-\left(\dfrac{1}{-2^2}\right):\dfrac{8^2}{4^4}\)
\(=\dfrac{2}{5}\cdot1+\dfrac{9}{10}\cdot\dfrac{2}{3}+\dfrac{1}{4}\cdot\dfrac{4^4}{8^2}\)
\(=\dfrac{2}{5}+\dfrac{18}{30}+\dfrac{4^3}{8^2}\)
\(=\dfrac{2}{5}+\dfrac{3}{5}+\dfrac{64}{64}=1+1=2\)
c: \(\dfrac{1}{3}+\dfrac{3}{7}+\dfrac{1}{7\cdot2}+\dfrac{5}{2\cdot13}+\dfrac{3}{13\cdot4}+\dfrac{5}{4\cdot21}\)
\(=\dfrac{1}{3}+\dfrac{3}{7}+\dfrac{1}{14}+\dfrac{5}{26}+\dfrac{3}{52}+\dfrac{5}{84}\)
\(=\left(\dfrac{3}{7}+\dfrac{1}{14}+\dfrac{5}{84}\right)+\left(\dfrac{1}{3}+\dfrac{5}{26}+\dfrac{3}{52}\right)\)
\(=\left(\dfrac{36}{84}+\dfrac{6}{84}+\dfrac{5}{84}\right)+\left(\dfrac{1}{3}+\dfrac{10}{52}+\dfrac{3}{52}\right)\)
\(=\dfrac{47}{84}+\dfrac{1}{3}+\dfrac{13}{52}\)
\(=\dfrac{47}{84}+\dfrac{28}{84}+\dfrac{1}{4}=\dfrac{75}{84}+\dfrac{21}{84}=\dfrac{96}{84}=\dfrac{8}{7}\)
