- Identify the distributive property: \(a(b+c) = a \cdot b + a \cdot c\).
- Apply the property to the given expression: \(8(3+4) = 8 \cdot 3 + 8 \cdot 4 = 24 + 8 \cdot 4\).
- Evaluate the multiplication to find the missing value: \(8 \cdot 4 = 32\).
- State the final missing value as \(\boxed{32}\).
Explanation
- Understanding the Distributive Property
The distributive property of multiplication over addition states that for any real numbers \(a, b,\) and \(c\), the following equality holds:
\[a(b+c) = a \cdot b + a \cdot c\]
In this specific problem, we are given the expression \(8(3+4)\). By comparing this to the general form \(a(b+c)\), we identify \(a = 8\), \(b = 3\), and \(c = 4\).
- Applying the Property
Applying the distributive property to the expression \(8(3+4)\), we distribute the \(8\) to both terms inside the parentheses:
\[8(3+4) = 8 \cdot 3 + 8 \cdot 4\]
We know that \(8 \cdot 3 = 24\). Substituting this into our equation, we get:
\[8(3+4) = 24 + 8 \cdot 4\]
The problem asks for the term that completes the equation \(8(3+4) = 24 + \dots\). By our calculation, the missing term is \(8 \cdot 4\).
- Final Calculation and Conclusion
Finally, we calculate the numerical value of the remaining term:
\[8 \cdot 4 = 32\]
Thus, the complete equation is:
\[8(3+4) = 24 + 32\]
The value that replaces the ellipsis is \(32\), which makes the total sum \(8 \cdot 7 = 56\), confirming that \(24 + 32 = 56\). Therefore, the missing value is \(\boxed{32}\).
Examples
The distributive property is a fundamental tool in algebra used to simplify expressions and solve equations. For example, if a store sells 8 packs of items, and each pack contains 3 blue pens and 4 red pens, you can calculate the total number of pens in two ways: either by adding the pens in one pack first and then multiplying by the number of packs, \(8 \times (3 + 4) = 8 \times 7 = 56\), or by calculating the total number of blue pens and red pens separately, \(8 \times 3 + 8 \times 4 = 24 + 32 = 56\). Both methods yield the same result, showcasing how the distributive property helps in breaking down complex calculations into manageable parts.