(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