(i)
\((\frac{5}{7}\times\frac{12}{13})\times\frac{7}{18}=\frac{5}{7}\times(\frac{12}{13}\times\frac{7}{18})\)
LHS = \((\frac{5}{7}\times\frac{12}{13})\times\frac{7}{18}\)
= \(\frac{5\times12}{7\times13}\times\frac{7}{18}\)
= \(\frac{60}{91}\times\frac{7}{18}\)
= \(\frac{60\times7}{91\times18}\)
= \(\frac{420}{1638}\)
In lowest terms,
\(\frac{420}{1638}= \frac{420 ÷42}{1638 ÷42}=\frac{10}{13}\)
RHS = \(\frac{5}{7}\times(\frac{12}{13}\times\frac{7}{18})\)
= \(\frac{5}{7}\times\frac{12\times7}{13\times18}\)
= \(\frac{5}{7}\times\frac{84}{234}\)
= \(\frac{420}{1638}\)
In lowest terms,
\(\frac{420}{1638}= \frac{420 ÷42}{1638 ÷42}=\frac{10}{13}\)
LHS=RHS
(ii)
\(\frac{-13}{24}\times(\frac{-12}{5}\times\frac{35}{36}=(\frac{-13}{24}\times\frac{-12}{5})\times\frac{35}{36}\)
LHS = \(\frac{-13}{24}\times(\frac{-12}{5}\times\frac{35}{36})\)
= \(\frac{-13}{24}\times\frac{-12\times35}{5\times36}\)
= \(\frac{-13}{24}\times\frac{-420}{180}\)
= \(\frac{60\times7}{24\times180}\)
= \(\frac{5460}{4320}\)
In lowest terms,
\(\frac{546}{432}= \frac{5460 ÷10}{4320 ÷10}=\frac{546}{432}\)
Further,
\(\frac{546}{432}= \frac{546 ÷6}{432 ÷6}=\frac{91}{72}\)
RHS = \((\frac{-13}{24}\times\frac{-12}{5})\times\frac{35}{36}\)
= \(\frac{-13\times-12}{24\times5}\times\frac{35}{36}\)
= \(\frac{156}{120}\times\frac{35}{36}\)
= \(\frac{5460}{4320}\)
In lowest terms,
\(\frac{546}{432}= \frac{5460 ÷10}{4320 ÷10}=\frac{546}{432}\)
Further,
\(\frac{546}{432}= \frac{546 ÷6}{432 ÷6}=\frac{91}{72}\)
LHS=RHS
(iii)
\((\frac{-9}{5}\times\frac{-10}{3}) \times\frac{21}{-4}=\frac{-9}{5}\times(\frac{-10}{3}\times\frac{21}{-4})\)
LHS = \((\frac{-9}{5}\times\frac{-10}{3})\times\frac{21}{-4}\)
= \(\frac{-9\times-10}{5\times3}\times\frac{21}{-4}\)
= \(\frac{90}{15}\times\frac{21}{-4}\)
= \(\frac{90\times21}{15\times-4}\)
= \(\frac{1890}{-60}= \frac{1890\times-1}{-60\times-1}=\frac{-1890}{6}\)
In lowest terms,
\(\frac{-1890}{60}= \frac{-1890\times-10}{60\times-10}=\frac{-189}{6}\)
RHS = \(\frac{-9}{5}\times(\frac{-10}{3}\times\frac{21}{-4})\)
= \(\frac{-9}{5}\times\frac{-10\times21}{3\times-4}\)
= \(\frac{-9}{5}\times\frac{-210}{-12}\)
= \(\frac{1890}{-60}= \frac{1890\times-1}{-60\times-1}=\frac{-1890}{60}\)
In lowest terms,
\(\frac{-1890}{60}= \frac{-1890\times-1}{-60\times-1}=\frac{-1890}{60}\)
Further,
\(\frac{-189}{6}= \frac{-189 ÷3}{6 ÷3}=\frac{-63}{2}\)
LHS=RHS