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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.