We start by solving the following expression:
[tex](-\frac{3}{4})(\frac{7}{8})[/tex]To multiply fractions we follow this rule:
[tex](\frac{a}{b})(\frac{c}{d})=\frac{a\times c}{b\times d}[/tex]So we do this with the expression:
[tex](-\frac{3}{4})(\frac{7}{8})=-\frac{3\times7}{4\times8}[/tex]Solving the operations:
[tex](-\frac{3}{4})(\frac{7}{8})=-\frac{21}{32}[/tex]Now we solve another of the expressions:
[tex](\frac{2}{3})(-4)(9)[/tex]In this case, we can solve the multiplication between 4 and 9 first:
[tex](\frac{2}{3})(-4)(9)=(\frac{2}{3})(-36)[/tex]And we consider that the rule to multiply a fraction by an integer is:
[tex](\frac{a}{b})(c)=\frac{a\times c}{b}[/tex]So in this case:
[tex](\frac{2}{3})(-4)(9)=\frac{2(-36)}{3}[/tex]And we solve the operations:
[tex]\begin{gathered} (\frac{2}{3})(-4)(9)=\frac{-72}{3} \\ (\frac{2}{3})(-4)(9)=-24 \end{gathered}[/tex]Next, we will solve the expression:
[tex](\frac{5}{16})(-2)(-4)(-\frac{4}{5})[/tex]In this case, we can also multiply the integers first, note that (-2)(-4) will be a positive number because we multiply two minus signs, so we get:
[tex](\frac{5}{16})(-2)(-4)(-\frac{4}{5})=(\frac{5}{16})(8)(-\frac{4}{5})[/tex]Now we multiply the first fraction by the integer following the same rule as in the last expression:
[tex]\begin{gathered} (\frac{5}{16})(-2)(-4)(-\frac{4}{5})=(\frac{5\times8}{16})(-\frac{4}{5}) \\ (\frac{5}{16})(-2)(-4)(-\frac{4}{5})=(\frac{40}{16})(-\frac{4}{5}) \end{gathered}[/tex]Now we multiply the fractions:
[tex]\begin{gathered} (\frac{5}{16})(-2)(-4)(-\frac{4}{5})=(\frac{40}{16})(-\frac{4}{5}) \\ (\frac{5}{16})(-2)(-4)(-\frac{4}{5})=-\frac{40\times4}{16\times5} \\ (\frac{5}{16})(-2)(-4)(-\frac{4}{5})=-\frac{160}{80} \end{gathered}[/tex]And the final result for this expression is:
[tex](\frac{5}{16})(-2)(-4)(-\frac{4}{5})=-2[/tex]Now we solve the last expression:
[tex](2\frac{3}{5})(\frac{7}{9})[/tex]in this expression, we have a mixed number which we need to convert to a fraction, the procedure to convert it is as follows:
[tex]2\frac{3}{5}=\frac{2\times5+3}{5}=\frac{13}{5}[/tex]We substitute this in our expression:
[tex](2\frac{3}{5})(\frac{7}{9})=(\frac{13}{5})(\frac{7}{9})[/tex]And multiply the fractions to get the final result:
[tex]\begin{gathered} (2\frac{3}{5})(\frac{7}{9})=\frac{13\times7}{5\times9} \\ (2\frac{3}{5})(\frac{7}{9})=\frac{91}{45} \end{gathered}[/tex]