- (a) \(5 \times 1 = 5\), using the Identity Property of Multiplication.
- (b) \(4 \times 9 = 9 \times 4\), using the Commutative Property of Multiplication.
- (c) \(3 \times (6 \times 7) = (3 \times 6) \times 7\), using the Associative Property of Multiplication.
- (d) \(0 \times 8 = 0\), using the Zero Property of Multiplication.
- The missing values are \(\boxed{1, 9, 7, 0}\).
Explanation
- Analysis of Multiplication Properties
To solve these problems, we need to apply the fundamental properties of multiplication. Each equation highlights a specific rule that governs how numbers behave under multiplication. We will evaluate each case individually to determine the missing value and identify the corresponding property.
- Solving the Equations and Identifying Properties
A\(5 \times \Box = 5\): Any number multiplied by 1 remains unchanged. Thus, \(5 \times 1 = 5\). This is known as the Identity Property of Multiplication, which states that for any number \(a\), \(a \times 1 = a\). Here, the missing value is \(1\).
B\(4 \times \Box = 9 \times 4\): The order of factors does not change the product. Therefore, \(4 \times 9 = 9 \times 4\). This is the Commutative Property of Multiplication, which states that \(a \times b = b \times a\). Here, the missing value is \(9\).
C\(3 \times (6 \times 7) = (3 \times 6) \times \Box\): The way in which factors are grouped does not change the product. Therefore, \(3 \times (6 \times 7) = (3 \times 6) \times 7\). This is the Associative Property of Multiplication, which states that \(a \times (b \times c) = (a \times b) \times c\). Here, the missing value is \(7\).
D\(\Box \times 8 = 0\): Any number multiplied by 0 results in 0. Therefore, \(0 \times 8 = 0\). This is the Zero Property of Multiplication, which states that for any number \(a\), \(a \times 0 = 0\). Here, the missing value is \(0\).
- Final Results
The completed equations and their corresponding properties are as follows:
A\(5 \times 1 = 5\) (Identity Property)
B\(4 \times 9 = 9 \times 4\) (Commutative Property)
C\(3 \times (6 \times 7) = (3 \times 6) \times 7\) (Associative Property)
D\(0 \times 8 = 0\) (Zero Property)
Thus, the missing values are \(1, 9, 7,\) and \(0\), respectively.
Examples
Understanding these properties helps in simplifying mathematical expressions efficiently. For instance, in a supermarket, if you buy 4 packs of juice, and each pack contains 9 juice boxes, knowing the Commutative Property allows you to quickly calculate the total as \(4 \times 9\) or \(9 \times 4\), both resulting in 36, depending on how you choose to group your mental calculation. These properties are fundamental in algebra for rearranging and solving equations more simply.