3. Distributivity of multiplication over addition for rational numbers

Theory:

The distributive property states that multiplying a sum by a number gives the same result as multiplying each added by the number and then adding the products together.
 
For any number \(a\), \(b\) and \(c\), we have \(a × b + c = a × b + a × c\).
 
Consider the number \(4\), \(3\) and \(2\) we have
 
\(4 × 3 + 2 = 4 × 3 + 4 × 2\). (LHS \(=\) RHS)
 
LHS: \(4 × 2 + 3 = 4 × 5\)
 
\(= 20\).
 
RHS: \(4 × 3 + 4 × 2\)

\(= 12 + 8\)
 
\(= 20\).
 
Thus, \(4 × 3 + 2 = 4 × 3 + 4 × 2\). (LHS \(=\) RHS)
 
For any three rational numbers ab, cd and ef, we have: ab \(×\) cd+ef \(=\) ab×cd \(+\) ab×ef.
 
Consider the rational numbers −34, 23 and  −56 we have
 
−34 \(×\) {23+ −56} \(=\) −34×23 \(+\)  −34× −56. (LHS \(=\) RHS)
 
LHS: −34 \(×\) {23+ −56}
 
−34 \(×\) {23+ −56} 
 
\(=\) −34 \(×\) {23×22+ −56} (LCM was taken to get the same denominator)
 
\(=\) −34 \(×\) {(4+ −5)6}
 
\(=\) −34 \(×\) −16 \(=\) 324 \(=\) 18.
 
RHS: −34×23 \(+\)  −34× −56:
 
−34×23 \(=\) −3×24×3 \(=\) −612 \(=\) −12.
 
And  −34× −56 \(=\) 58.
 
Therefore  −34×23 \(+\)  −34× −56
 
 \(=\) −12 \(+\) 58 
 
\(=\)−12×44 \(+\) 58×11 (LCM was taken to get the same denominator)
 
\(=\) ( −4+5)8
 
\(=\) 18.  
 
Thus,  −34 \(×\) {23+ −56} \(=\) −34×23 \(+\)  −34× −56. (LHS \(=\) RHS)
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