Using the Distributive Property

The distributive property is a characteristic of numbers that involves both addition and multiplication. It is used often in algebra, and we can use it now to obtain exact results for a multiplication.

Suppose we wish to compute \( 3(2+5) \). We can proceed in either of two ways, one way which is known to us already (the order of operations), and a new way (the distributive property).

Compute \( 3(2+5) \) using the order of operations.

\( 3(2+5) \)

Operate inside the parentheses first: \( 2+5 = 7 \).

\( 3(2+5) = 3⋅7 \)

Now multiply 3 and 7.

\( 3(2+5) = 3⋅7 = 21 \)

Thus \( 3(2+5) = 21 \).

Compute \( 3(2+5) \) using the distributive property.

We know that multiplication describes repeated addition. Thus,

\( \begin{aligned} 3(2+5) &=\underbrace{2+5+2+5+2+5}_{2+5 \text { appears } 3 \text { times }} & & \\ &=2+2+2+5+5+5 & & \text { (by the commutative property of addition) } \\ &=3 \cdot 2+3 \cdot 5 & & \text { (since multiplication describes repeated addition) } \\ &=6+15 & & \\ &=21 & & \end{aligned} \)

Thus, \( 3(2+5) = 21\).

Let's look again at this use of the distributive property.

\( \begin{array}{l} 3(2+5)=\underbrace{2+5+2+5+2+5}_{2+5 \text { appears } 3 \text { times }}\\ 3(2+5)=\underbrace{2+2+2}_{2 \text { appears } 3 \text { times }}+\underbrace{5+5+5}_{5 \text { appears } 3 \text { times }}\\ \begin{array}{l} 3(2+5)=3 \cdot 2+3 \cdot 5 \\  \quad \qquad \qquad 3 \times 2 \quad 3 \times 5 \end{array} \end{array} \)

The 3 has been distributed to the 2 and 5.

This is the distributive property. We distribute the factor to each addend in the parentheses. The distributive property works for both sums and differences.

Sample Set A

Using the order of operations, we get

\( \begin{aligned} 4(6+2) =4 \cdot 8 \\ =32 \end{aligned} \)

Using the order of operations, we get

\( \begin{aligned} 8(9+6) &=8 \cdot 15 \\ &=120 \end{aligned} \)

Source: Rice University, https://cnx.org/contents/XeVIW7Iw@4.6:Gmai8dss@2/Mental-Arithmetic-Using-the-Distributive-Property
This work is licensed under a Creative Commons Attribution 4.0 License.

Use the distributive property to compute each value.

1. \( 6 (8 + 4)\)
   
   
2. \( 4 (4 + 7) \)
   
   
3. \(8 (2 + 9)\)
   
   
4. \( 12 (10 + 3)\)
   
   
5. \( 6 (11 − 3) \)
   
   
6. \( 8 (9 − 7) \)
   
   
7. \( 15 (30 − 8) \)

1. \( 6⋅8 + 6⋅4 = 48 + 24 = 72 \)
   
   
2. \( 4⋅4 + 4⋅7 = 16 + 28 = 44 \)
   
   
3. \( 8⋅2 + 8⋅9 = 16 + 72 = 88 \)
   
   
4. \( 12⋅10 + 12⋅3 = 120 + 36 = 156 \)
   
   
5. \( 6⋅11 − 6⋅3 = 66 − 18 = 48 \)
   
   
6. \( 8⋅9 – 8⋅7 = 72 – 56 = 16 \)
   
   
7. \( 15⋅30 − 15⋅8 = 450 − 120 = 330 \)