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[tex] \bf{ \underline{ \underline{R equired \: E xplanation..}}}[/tex]
Given to verify that :-
a × ( b + c ) = ( a × b ) + ( a× c) where,
Take the L.H.S :-
a × ( b + c ) =
[tex] \rm \: \dfrac{ - 15}{4} \times \bigg( \dfrac{4}{7} - \dfrac{12}{5} \bigg)[/tex]
L.C.M of 7,5 is 35 .
[tex] \longrightarrow \rm \: \dfrac{ - 15}{4} \times \bigg( \dfrac{4 \times 5 - 12 \times 7}{35} \bigg)[/tex]
[tex] \longrightarrow \rm \: \dfrac{ - 15}{4} \times \bigg( \dfrac{20 - 84}{35} \bigg)[/tex]
[tex] \longrightarrow \rm \: \dfrac{ - 15}{4} \times \bigg( \dfrac{ - 64}{35} \bigg)[/tex]
[tex] \longrightarrow \bigg( \rm \: \dfrac{ - 15 \times - 64}{4 \times 35} \bigg)[/tex]
[tex] \longrightarrow \bigg( \rm \: \dfrac{ 15 \times 64}{4 \times 35} \bigg)[/tex]
[tex] \longrightarrow \bigg( \rm \: \dfrac{ 15 \times 16}{ 35} \bigg)[/tex]
[tex] \longrightarrow \bigg( \rm \: \dfrac{ 3 \times 16}{ 7} \bigg)[/tex]
[tex] \longrightarrow \bigg( \rm \: \dfrac{ 48}{ 7} \bigg)[/tex]
L.H.S = 48/7
___________________...
Take R.H.S:-
( a × b ) + (a ×c )
[tex] \longrightarrow \bigg( \rm \dfrac{ - 15}{4} \times \dfrac{4}{7} \bigg) + \bigg( \dfrac{ - 15}{4} \times \dfrac{ - 12}{5} \bigg)[/tex]
[tex] \longrightarrow \bigg( \rm \dfrac{ - 15}{ \not4} \times \dfrac{ \not4}{7} \bigg) + \bigg( \dfrac{ \not- 15}{ \not4} \times \dfrac{ \not - 12}{ \not5} \bigg)[/tex]
[tex] \longrightarrow \bigg( \rm\dfrac{ - 15}{7} \bigg) + \bigg( 3 \times3 \bigg)[/tex]
[tex] \longrightarrow \bigg( \rm\dfrac{ - 15}{7} \bigg) + \bigg( 9 \bigg)[/tex]
[tex] \longrightarrow \bigg( \rm\dfrac{ - 15 + 63}{7} \bigg) [/tex]
[tex] \longrightarrow \bigg( \rm\dfrac{ 48 }{7} \bigg) [/tex]
So,
R.H.S =48/7
__________________...
L.H.S = R.H.S = 48/7
Hence verified!