Distributive Property of Multiplication over Addition

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 product of a number with the sum of two other numbers is equivalent to multiplying that number by each addend individually and then adding the results 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!