(i)
\(\frac{3}{7}\times(\frac{5}{6}+\frac{12}{13})=(\frac{3}{7}\times\frac{5}{6})+(\frac{3}{7}\times\frac{12}{13})\)
LHS = \(\frac{3}{7}\times(\frac{5}{6}+\frac{12}{13})\)
\(=\frac{3}{7}\times(\frac{5\times13+12\times6}{78})\)
= \(\frac{3}{7}\times(\frac{65+72}{78})\)
= \(\frac{3}{7}\times(\frac{137}{78})\)
= \(\frac{3\times137}{7\times78}\)
= \(\frac{411}{546}\)
In lowest terms,
\(\frac{411}{546}= \frac{411\div3}{546\div3}=\frac{137}{182}\)
RHS = \((\frac{3}{7}\times\frac{5}{6})+(\frac{3}{7}\times\frac{12}{13})\)
= \((\frac{3\times5}{7\times6})+(\frac{3\times12}{7\times13})\)
= \(\frac{15}{42}+(\frac{36}{91})\)
= \(\frac{15\times13+36\times6}{546}\)
= \(\frac{195+216}{546}\)
= \(\frac{411}{546}\)
In lowest terms,
\(\frac{411}{546}= \frac{411\div3}{546\div3}=\frac{137}{182}\)
LHS=RHS
(ii)
\(\frac{-15}{4}\times(\frac{3}{7}+\frac{-12}{5}) =(\frac{-15}{4}\times\frac{3}{7})+(\frac{-15}{4}\times\frac{-12}{5})\)
LHS = \(\frac{-15}{4}\times(\frac{3}{7}+\frac{-12}{5})\)
= \(\frac{-15}{4}\) \(\times(\frac{3\times5+(-12)\times7}{35})\)
= \(\frac{-15}{4}\times(\frac{15-84}{35})\)
= \(\frac{-15}{4}\times(\frac{-69}{35})\)
= \((\frac{-15\times-69}{4\times35})\)
= \(\frac{1035}{140}\)
In lowest terms,
\(\frac{1035}{140}=\frac{1035\div5}{140\div5} =\frac{207}{28}\)
RHS = \((\frac{-15}{4}\times\frac{3}{7})+(\frac{-15}{4}\times\frac{-12}{5})\)
= \((\frac{-15\times3}{4\times7})+(\frac{-1500}{4}\times\frac{-12}{5})\)
= \(\frac{-45}{28}+(\frac{180}{20})\)
= \(\frac{-45\times5+180\times7}{140}\)
= \(\frac{-225+1260}{140}\)
= \(\frac{1035}{140}\)
In lowest terms,
\(\frac{1035}{140}=\frac{1035\div5}{140\div5} =\frac{207}{28}\)
LHS=RHS
(iii)
\((\frac{-8}{3}+\frac{-13}{12})\times\frac{5}{6} =(\frac{-8}{3}\times\frac{5}{6})+(\frac{-13}{12}\times\frac{5}{6})\)
LHS = \((\frac{-8}{3}+\frac{-13}{12})\times\frac{5}{6}\)
= \((\frac{-8\times4+(-13)\times1}{12})\times\frac{5}{6}\)
= \((\frac{-32-13}{12})\times(\frac{5}{6})\)
= \(\frac{-45}{12}\times\frac{5}{6}\)
= \(\frac{-45\times5}{12\times6}\)
= \(\frac{-225}{72}\)
In lowest terms,
\(\frac{-225}{72}= \frac{-225\div9}{72\div9}=\frac{-25}{8}\)
RHS = \((\frac{-8}{3}\times\frac{5}{6})+(\frac{-13}{12}\times\frac{5}{6})\)
= \((\frac{-8\times5}{3\times6})+(\frac{-13\times5}{12\times6})\)
= \(\frac{-40}{18}+(\frac{-65}{72})\)
= \(\frac{-40\times4+(-65)\times1}{72}\)
= \(\frac{-160-65}{72}\)
= \(\frac{-225}{72}\)
In lowest terms,
\(\frac{-225}{72}= \frac{-225\div9}{72\div9}=\frac{-25}{8}\)
LHS=RHS
(iv)
\(\frac{-16}{7}\times(\frac{-8}{9}+\frac{-7}{6}) =(\frac{-16}{7}\times\frac{-7}{6})\)
LHS = \(\frac{-16}{7}\times(\frac{-8}{9}+\frac{-7}{6})\)
= \(\frac{-16}{7}\times(\frac{-8\times2+(-7)\times3}{18})\)
= \(\frac{-16}{7}\times(\frac{-16-21}{18})\)
= \(\frac{-16}{7}(\frac{-37}{18})\)
= \(\frac{-16\times-37}{7\times18}\)
= \(\frac{592}{126}\)
In lowest terms,
\(\frac{592}{126}=\frac{592\div2}{126\div2}=\frac{296}{63}\)
RHS = \((\frac{-16}{7}\times\frac{-8}{9})+(\frac{-16}{7}\times\frac{-7}{6})\)
= \((\frac{-16\times-18}{7\times9})+ (\frac{-16\times-7}{7\times6})\)
= \(\frac{128}{63}+(\frac{112}{42})\)
= \(\frac{128\times2+112\times3}{126}\)
= \(\frac{592}{126}\)
In lowest terms,
\(\frac{592}{126}=\frac{592\div2}{126\div2}=\frac{296}{63}\)
LHS=RHS