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 3423 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)