1: \(A=4\frac{8}{23}+5\frac78-9\frac{8}{23}\)

\(=4+\frac{8}{23}+5+\frac78-9-\frac{8}{23}\)

\(=\left(4+5-9\right)+\left(\frac{8}{23}-\frac{8}{23}\right)+\frac78=\frac78\)

2: \(B=6\frac38+5\frac12\)

\(=6+5+\frac38+\frac12\)

\(=11+\frac78=\frac{11\cdot8+7}{8}=\frac{95}{8}\)

3: \(C=\frac{-3}{7}\cdot\frac59+\frac49\cdot\frac{-3}{7}\)

\(=-\frac37\left(\frac59+\frac49\right)=-\frac37\cdot1=-\frac37\)

4: \(D=8\frac27-\left(3\frac49+4\frac27\right)\)

\(=8+\frac27-3-\frac49-4-\frac27\)

\(=1-\frac49=\frac59\)

5: \(7\frac47-\left(2\frac13+3\frac47\right)\)

\(=7+\frac47-2-\frac13-3-\frac47\)

\(=2-\frac13=\frac53\)

6: \(-\frac{5}{23}\cdot\frac{17}{26}+\frac{5}{23}\cdot\frac{-9}{26}\)

\(=-\frac{5}{23}\left(\frac{17}{26}+\frac{9}{26}\right)=-\frac{5}{23}\cdot1=-\frac{5}{23}\)

7: \(\left(\frac37-\frac{8}{13}\right):\frac{-3}{4}+\left(\frac{11}{7}-\frac{5}{13}\right):\frac{-3}{4}\)

\(=\left(\frac37-\frac{8}{13}+\frac{11}{7}-\frac{5}{13}\right):\frac{-3}{4}\)

\(=\left(2-1\right)\cdot\frac{-4}{3}=-\frac43\)

8: \(-\frac37+\frac{15}{26}-\left(\frac{2}{13}-\frac37\right)\)

\(=-\frac37+\frac{15}{26}-\frac{2}{13}+\frac37=\frac{15}{26}-\frac{2}{13}=\frac{11}{26}\)

9: \(3\frac57-\frac57\cdot\frac{4}{15}+\frac{-5}{7}\cdot\frac{11}{15}\)

\(=3+\frac57-\frac57\left(\frac{4}{15}+\frac{11}{15}\right)\)

\(=3+\frac57-\frac57\)

=3

10: \(\frac12+\frac{-1}{5}+\frac{-5}{7}+\frac16+\frac{-3}{35}+\frac13+\frac{1}{41}\)

\(=\left(\frac12+\frac13+\frac16\right)-\left(\frac15+\frac57+\frac{3}{35}\right)+\frac{1}{41}\)

\(=\frac{3+2+1}{6}-\frac{7+25+3}{35}+\frac{1}{41}=\frac{1}{41}\)

11: \(\frac{2^2}{3}\cdot\frac{3^2}{8}\cdot\frac{4^2}{15}\cdot\frac{5^2}{24}\cdot\frac{6^2}{35}\cdot\frac{7^2}{48}\cdot\frac{8^2}{63}\cdot\frac{9^2}{80}\)

\(=\frac{2^2}{1\cdot3}\cdot\frac{3^2}{2\cdot4}\cdot\ldots\cdot\frac{9^2}{8\cdot10}\)

\(=\frac{2\cdot3\cdot\ldots\cdot9}{1\cdot2\cdot\ldots\cdot8}\cdot\frac{2\cdot3\cdot\ldots\cdot9}{3\cdot4\cdot\ldots\cdot10}=\frac91\cdot\frac{2}{10}=\frac95\)

13: \(\frac47+\frac34+\frac27+\frac54+\frac17\)

\(=\left(\frac47+\frac27+\frac17\right)+\left(\frac34+\frac54\right)\)

=1+2

=3

14: \(\frac{-4}{12}+\frac{18}{45}+\frac{-6}{9}+\frac{-21}{35}+\frac{6}{30}\)

\(=-\frac23-\frac13+\frac25+\frac15-\frac35\)

=-1+0

=-1

15: \(\frac{31}{17}+\frac{-5}{13}+\frac{-8}{13}-\frac{14}{17}\)

\(=\left(\frac{31}{17}-\frac{14}{17}\right)-\left(\frac{5}{13}+\frac{8}{13}\right)\)

=1-1

=0

16: \(\frac{-5}{7}\cdot\frac{2}{11}+\frac{-5}{7}\cdot\frac{9}{11}+\frac57\)

\(=-\frac57\left(\frac{2}{11}+\frac{9}{11}\right)+\frac57\)

\(=-\frac57+\frac57=0\)