For any three real numbers \(a\), \(b\), and \(c\)
$$a\left( {b + c} \right) = \left( {a \times b} \right) + \left( {a \times c} \right)$$
The property tells us that we get the same answer if we multiply an outer number by the sum of two numbers in expanded form or when we first multiply the outer number to each of the addends inside the parenthesis then add them together.
If we observe the "formula", the outer number \(a\) is literally being distributed into the inner numbers ( \(b\) and \(c\) ) separated by the addition operation.
Let's go over a quick example to illustrate the property.
Is the statement below true?
$$3\left( {4 + 5} \right) = \left( {3 \times 4} \right) + \left( {3 \times 5} \right)$$
Simplifying the left-hand side (LHS):
$$\eqalign{
3\left( {4 + 5} \right) &= 3\left( 9 \right) \cr
&= \boxed{27}\cr} $$
Simplifying the right-hand side (RHS):
$$\eqalign{
\left( {3 \times 4} \right) + \left( {3 \times 5} \right) &= 12 + 15 \cr
&=\boxed{ 27} \cr} $$
It sure does! The LHS equals the RHS.
Let's have another example!
Rewrite using the Distributive Property then simplify
$$7\left( {3 + 8} \right)$$
Here we go.
$$\eqalign{
7\left( {3 + 8} \right) &= \left( {7 \times 3} \right) + \left( {7 \times 8} \right) \cr
&= 21 + 56 \cr
&= 77 \cr} $$
We can verify if our answer is right by simplifying the given problem without the distributive property.
$$\eqalign{
7\left( {3 + 8} \right) &= 7\left( {11} \right) \cr
&= 77 \cr} $$
It checks!