$a(b+c)=a \cdot b+a \cdot c$, where $a, b$, and $c$ are real numbers. Use the distributive property to simplify the expression. $8(3+4)=24+\square$
Answer (1)
Apply the distributive property: 8 ( 3 + 4 ) = 8 × 3 + 8 × 4 .
Calculate the first term: 8 × 3 = 24 .
Calculate the second term: 8 × 4 = 32 .
The missing value is 32 .
Explanation
Understanding the Problem We are given the equation 8 ( 3 + 4 ) = 24 + □ . We need to find the missing value using the distributive property. The distributive property states that a ( b + c ) = a ⋅ b + a ⋅ c for real numbers a , b , c .
Applying Distributive Property Applying the distributive property to the left side of the equation, we have 8 ( 3 + 4 ) = 8 ⋅ 3 + 8 ⋅ 4 .
Calculating First Term We know that 8 ⋅ 3 = 24 , so we can rewrite the equation as 8 ( 3 + 4 ) = 24 + 8 ⋅ 4 .
Calculating Second Term Now we need to calculate 8 ⋅ 4 . The result of this operation is 32 .
Finding the Missing Value Substituting this value back into the equation, we get 8 ( 3 + 4 ) = 24 + 32 . Therefore, the missing value is 32.
Examples
The distributive property is useful in everyday situations, such as calculating the total cost of multiple items. For example, if you buy 8 items that each cost $3 plus an additional $4 for tax, the total cost can be calculated as 8 * (3 + 4). Using the distributive property, this is the same as (8 * 3) + (8 * 4), which breaks down the calculation into simpler steps. This makes it easier to mentally calculate the total cost: $24 for the items plus $32 for the tax, totaling $56.
