Fill in the blanks. Then, choose the property of multiplication you used.

(a) $3 \times(2 \times 8)=(3 \times 2) \times \square$
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\hline (b) $6 \times 9=9 \times$ □ &

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\hline (c) $1 \times \square=5$ &

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\hline (d) $5 \times 0=$ □ &

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We are given four equations that demonstrate fundamental properties of multiplication. Our task is to identify the missing value for each and determine the specific property of multiplication applied in each case. The properties we are considering are the associative, commutative, multiplicative identity, and zero properties. We will examine each equation individually to determine the correct missing value and the corresponding rule that makes the equation true.

Solving Equation (a)

Equation (a) is $3 \times (2 \times 8) = (3 \times 2) \times \square$. This equation illustrates the associative property of multiplication, which states that the grouping of factors does not change the product, i.e., $a \times (b \times c) = (a \times b) \times c$. By comparing the sides, we can see that the missing factor must be $8$. Thus, $3 \times (2 \times 8) = (3 \times 2) \times 8$.

Solving Equation (b)

Equation (b) is $6 \times 9 = 9 \times \square$. This demonstrates the commutative property of multiplication, which states that the order of the factors does not change the product, i.e., $a \times b = b \times a$. Since we are multiplying $6$ and $9$, the missing factor on the right side must be $6$. Thus, $6 \times 9 = 9 \times 6$.

Solving Equation (c)

Equation (c) is $1 \times \square = 5$. This demonstrates the multiplicative identity property, which states that the product of any number and $1$ is the number itself, i.e., $1 \times a = a$. Here, since the result is $5$, the missing factor must be $5$. Thus, $1 \times 5 = 5$.

Solving Equation (d)

Equation (d) is $5 \times 0 = \square$. This demonstrates the zero property of multiplication, which states that the product of any number and $0$ is $0$, i.e., $a \times 0 = 0$. Therefore, the result of $5 \times 0$ is $0$. Thus, $5 \times 0 = 0$.

Examples

Understanding properties of multiplication is essential for efficient mental math and simplifying complex expressions in everyday life. For instance, if you are calculating the total cost of items during a shopping trip, you can use the commutative property to reorder numbers for easier multiplication, or the associative property to group items that are easier to multiply together first, like multiplying by 10s or 5s. These simple rules help make calculations quicker and less prone to errors when managing your daily tasks, such as budgeting or inventory planning.

(a) $3 \times(2 \times 8)=(3 \times 2) \times \square$	\begin{tabular}[t]{l} □
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