State the properties illustrated by these statement:$( i)$. $\frac{5}{6} \times 1=1 \times \frac{5}{6}$$( ii)$. $(\frac{-3}{7}) \times \frac{5}{8}=\frac{5}{8} \times(\frac{-3}{7})$$( iii)$. $\frac{-6}{11} \times( \frac{-1}{-18})=( \frac{-1}{-18}) \times \frac{-6}{11}$$( iv)$. $\frac{-4}{5} \times(\frac{5}{12}+\frac{7}{18})=\frac{-4}{5} \times \frac{5}{12}+\frac{-4}{5} \times \frac{7}{18}$$( v)$. $[\frac{3}{8} \times(\frac{-9}{16})] \times \frac{5}{6}=\frac{3}{8} \times[( \frac{-9}{16}) \times \frac{5}{6}]$$( vi)$. $\frac{8}{-13} \times(\frac{-13}{8})=1$$( vii)$. $\frac{12}{19} \times[\frac{-3}{18} \times(\frac{12}{-33})]=[\frac{12}{19} \times( \frac{-3}{18})] \times( \frac{12}{-33})$$( viii)$. $\frac{1}{3} \times(\frac{-7}{11}-\frac{2}{5})=\frac{1}{3} \times(\frac{-7}{11})-\frac{1}{3} \times \frac{2}{5}$

Given: $( i)$. $\frac{5}{6} \times 1=1 \times \frac{5}{6}$

$( ii)$. $(\frac{-3}{7}) \times \frac{5}{8}=\frac{5}{8} \times(\frac{-3}{7})$

$( iii)$. $\frac{-6}{11} \times( \frac{-1}{-18})=( \frac{-1}{-18}) \times \frac{-6}{11}$

$( iv)$. $\frac{-4}{5} \times(\frac{5}{12}+\frac{7}{18})=\frac{-4}{5} \times \frac{5}{12}+\frac{-4}{5} \times \frac{7}{18}$

$( v)$. $[\frac{3}{8} \times(\frac{-9}{16})] \times \frac{5}{6}=\frac{3}{8} \times[( \frac{-9}{16}) \times \frac{5}{6}]$

$( vi)$. $\frac{8}{-13} \times(\frac{-13}{8})=1$

$( vii)$. $\frac{12}{19} \times[\frac{-3}{18} \times(\frac{12}{-33})]=[\frac{12}{19} \times( \frac{-3}{18})] \times( \frac{12}{-33})$

$( viii)$. $\frac{1}{3} \times(\frac{-7}{11}-\frac{2}{5})=\frac{1}{3} \times(\frac{-7}{11})-\frac{1}{3} \times \frac{2}{5}$

To do: To state the property.

Solution: 

$( i)$. $\frac{5}{6} \times 1=1 \times \frac{5}{6}$

$\because$ here $a\times b=b\times a$. Thus it is a commutative property.

$( ii)$. $(\frac{-3}{7}) \times \frac{5}{8}=\frac{5}{8} \times(\frac{-3}{7})$.

 

$\because$ here $a\times b=b\times a$. Thus it is a commutative property.

$( iii)$. $\frac{-6}{11} \times( \frac{-1}{-18})=( \frac{-1}{-18}) \times \frac{-6}{11}$

$\because$ here $a\times b=b\times a$. Thus it is a commutative property.

$( iv)$. $\frac{-4}{5} \times(\frac{5}{12}+\frac{7}{18})=\frac{-4}{5} \times \frac{5}{12}+\frac{-4}{5} \times \frac{7}{18}$

$\because a( b+c)=ab+ac$, thus it is distributive property of multiplication.

$( v)$. $[\frac{3}{8} \times(\frac{-9}{16})] \times \frac{5}{6}=\frac{3}{8} \times[( \frac{-9}{16}) \times \frac{5}{6}]$

$\because$ Here $a\times( b\times c)=( a\times  b)\times c$, thus it is associative property.

$( vi)$. $\frac{8}{-13} \times(\frac{-13}{8})=1$

$\because$ Here $a\times b=1$, thus it is a multiplicative inverse property.

$( vii)$. $\frac{12}{19} \times[\frac{-3}{18} \times(\frac{12}{-33})]=[\frac{12}{19} \times( \frac{-3}{18})] \times( \frac{12}{-33})$

$\because$ Here $a\times( b\times c)=( a\times  b)\times c$, thus it is associative property.

$( viii)$. $\frac{1}{3} \times(\frac{-7}{11}-\frac{2}{5})=\frac{1}{3} \times(\frac{-7}{11})-\frac{1}{3} \times \frac{2}{5}$

$\because a( b-c)=ab-ac$, Thus it distributive property.