Explainer: Evaluating Numerical Expressions: Distributive Property

In this explainer, we will learn how to use the distributive property of multiplication for addition and subtraction to rewrite numerical expressions and evaluate them.

Definition: The Distributive Property

The distributive property states that any expression in the form 𝑎(𝑏+𝑐) is the same as 𝑎𝑏+𝑎𝑐; that is, 𝑎(𝑏+𝑐)=𝑎𝑏+𝑎𝑐.

The distributive property can be applied to both numeric and algebraic expressions. Here we are going to look at using the property on numeric expressions, that is, where 𝑎, 𝑏, and 𝑐 are numbers. For example, if we consider the expression 3(4+2), the distributive property tells us that this is equivalent to 3(4)+3(2).

We can represent the distributive property diagrammatically using an area model. Consider the shown rectangle.

The rectangle is formed of two smaller rectangles with equal width, one with length 4, and the other with length 2. The area of the blue rectangle is 3×4 and the area of the red rectangle is 3×2. Combining these two expressions, we get that the total area of the rectangle is 3(4)+3(2).

We can also calculate the area directly by finding the total length and multiplying by the width: 3(4+2).

These two expressions are equivalent, which confirms that 3(4+2)=3(4)+3(2).

With this in mind, let us look at some examples where we rewrite expressions using the distributive property.

Example 1: Using the Distributive Property on Numerical Expressions Containing a Sum

Rewrite the expression 19(34+21) using the distributive property.

Answer

Recall the distributive property which states that 𝑎(𝑏+𝑐)=𝑎𝑏+𝑎𝑐. Note here that the expression contains a. If we apply this to the given expression, we have that 𝑎=19, 𝑏=34, and 𝑐=21, which tells us that 19(34+21)=19(34)+19(21).

Remember that between each number and parenthesis is a multiplication operation; we can write our expression as 19×34+19×21.

Example 2: Using the Distributive Property on Numerical Expressions Containing a Difference

Rewrite the expression 20(113) using the distributive property.

Answer

Recall the distributive property which states that 𝑎(𝑏+𝑐)=𝑎𝑏+𝑎𝑐. Note here that the expression contains a negative, but we can still apply the distributive property as we can rewrite the expression as 20(11+(3)).

If we apply this to the given expression, we have that 𝑎=20, 𝑏=11, and 𝑐=3, which tells us that 20(113)=20(11)+20(3).

Simplifying (remember that between each number and parenthesis is a multiplication operation), we can write our expression as 20×1120×3.

Note here that we could have used an adapted version of the distributive property which is 𝑎(𝑏𝑐)=𝑎(𝑏)𝑎(𝑐), which you may find easier to use.

We can use the distributive property to evaluate expressions in different ways. One way is generally a more efficient approach than the other, but we will demonstrate both.

Example 3: Using the Distributive Property to Expand a Numerical Expression

Evaluate 3(26+11) using the distributive property.

Answer

Recall the distributive property which states that 𝑎(𝑏+𝑐)=𝑎𝑏+𝑎𝑐. This tells us that we can rewrite the expression as 3(26)+3(11).

We can then evaluate each part of the sum to get 78+33=111.

Example 4: Using the Distributive Property to Factor a Numerical Expression

Evaluate 5(40)+5(26) using the distributive property.

Answer

Recall the distributive property which states that 𝑎(𝑏+𝑐)=𝑎𝑏+𝑎𝑐 and equally that 𝑎𝑏+𝑎𝑐=𝑎(𝑏+𝑐). We can use this identity to rewrite our expression as 5(40+26), which can then be simplified to 5(66)=330.

Example 4 is generally a more efficient approach when evaluating expressions; however, it is important to see how both approaches work.

We can also take a similar approach when evaluating expressions in the form 𝑎(𝑏𝑐). This expression can be rewritten as 𝑎(𝑏+(𝑐)), and then the distributive property tells us that this is equivalent to 𝑎(𝑏)+𝑎(𝑐)=𝑎(𝑏)𝑎(𝑐). Therefore, we have that 𝑎(𝑏𝑐)=𝑎(𝑏)𝑎(𝑐).

We will now look at an example that demonstrates using this adapted form of the distributive property.

Example 5: Using the Distributive Property to Factor a Numerical Expression

Using the distributive property, evaluate 5(43)5(10).

Answer

Recall the distributive property which states that 𝑎(𝑏+𝑐)=𝑎𝑏+𝑎𝑐.

Similarly, we have that 𝑎(𝑏𝑐)=𝑎𝑏𝑎𝑐.

We can then use this identity to rewrite our expression as 5(4310).

This simplifies to the product 5(33), which is equal to 165.

Key Points

  1. The distributive property states that an expression in the form 𝑎(𝑏+𝑐) is equivalent to 𝑎(𝑏)+𝑎(𝑐).
  2. Similarly, if the operation is negative, we have that 𝑎(𝑏𝑐)=𝑎(𝑏)𝑎(𝑐) (this is just an extension of the distributive rule stated above when 𝑐 is negative).
  3. The distributive property can be used to simplify calculations with numerical expressions (particularly in the case of negatives).

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