(i)
\((\frac{3}{4}+\frac{-2}{5})+\frac{-7}{10}= \frac{3}{4}+ (\frac{-2}{5}+\frac{-7}{10})\)
LHS = \((\frac{3}{4}+\frac{-2}{5})\) \(+\frac{-7}{10}\)
= \((\frac{3\times5+(-2)\times4)}{20})+\frac{-7}{10}\)
= \((\frac{15-8}{20}) + \frac{-7}{10}\)
= \(\frac{7}{20}+\frac{-7}{10}\)
= \(\frac{7\times1+(-7)\times2}{20}\)
= \(\frac{7-14}{20}\)
= \(\frac{-7}{20}\)
RHS = \(\frac{3}{4}+(\frac{-2}{5}+\frac{-7}{10})\)
= \(\frac{3}{4}+ (\frac{(-2)\times2+(-7)\times1}{10})\)
= \(\frac{3}{4}\) + \((\frac{-4-7}{10})\)
= \(\frac{3}{4}+\frac{-11}{10}\)
= \(\frac{3\times5+(-11)\times2}{20}\)
= \(\frac{15-22}{20}\)
= \(\frac{-7}{20}\)
RHS = LHS
(ii)
\((\frac{-7}{11}+\frac{2}{-5})+\frac{-13}{22}= \frac{-7}{11}+ (\frac{2}{-5}+\frac{-13}{22})\)
LHS = \((\frac{-7}{11}+\frac{2}{-5})\) \(+\frac{-13}{22}\)
= \((\frac{-7}{11}+\frac{2}{-5}) +\frac{-13}{22}\)
= \((\frac{-7\times5+(-2)\times11)}{55})+\frac{-13}{22}\)
= \((\frac{-35-22}{55}) +\frac{-13}{22}\)
= \(\frac{-57}{55}+\frac{-13}{22}\)
= \(\frac{-57\times2+(-13)\times5}{110}\)
= \(\frac{-144-65}{110}\)
= \(\frac{-179}{110}\)
RHS = \(\frac{-7}{11}+(\frac{2}{-5}+\frac{-13}{22})\)
= \(\frac{-7}{11}+(\frac{-2}{5}+\frac{-13}{22})\)
= \(\frac{-7}{11}+(\frac{(-2)\times22+(-13)\times5}{-110})\)
= \(\frac{-7}{11}+(\frac{-44-65}{-110})\)
= \(\frac{-7}{11}+\frac{-109}{110}\)
= \(\frac{-7\times10+(-109)\times1}{110}\)
= \(\frac{-70-109}{20}\)
= \(\frac{-179}{110}\)
RHS = LHS
(iii)
\(-1+(\frac{-2}{3}+\frac{-3}{4}) = (-1 + \frac{-2}{3}+\frac{-3}{4})\)
LHS = \(-1+(\frac{-2}{3}+\frac{-3}{4})\)
= \(\frac{-1}{1} +(\frac{-2}{3}+\frac{-3}{4})\)
= \(\frac{-1}{1}+(\frac{-2\times4+(-3)\times3}{12})\)
= \(\frac{-1}{1}+(\frac{-8-9}{12})\)
= \(\frac{-1}{1}+\frac{-17}{12}\)
= \(\frac{-1\times12+(-17)\times1}{12}\)
= \(\frac{-12-17}{12}\)
= \(\frac{-29}{12}\)
RHS = \((-1 +\frac{-2}{3}) +\frac{-3}{4}\)
= \((\frac{-1}{1}+\frac{-2}{3})+\frac{-3}{4}\)
= \((\frac{-1\times3+(-2)\times1}{3})+\frac{-3}{4}\)
= \((\frac{-3-2}{3})+\frac{-3}{4}\)
= \(\frac{-5}{3}+\frac{-3}{4}\)
= \(\frac{-5\times4+(-3)\times3}{12}\)
= \(\frac{-20-9}{20}\)
= \(\frac{-29}{12}\)
RHS = LHS