- Part (a): \(7 \times 0 = \boxed{0}\), Zero Property.
- Part (b): \(\boxed{1} \times 8 = 8\), Identity Property.
- Part (c): \(\boxed{2} \times (3 \times 4) = (2 \times 3) \times \boxed{4}\), Associative Property.
- Part (d): \(9 \times 8 = 8 \times \boxed{9}\), Commutative Property.
Explanation
- Problem Analysis
The goal is to solve four multiplication problems by identifying the missing factor and naming the mathematical property applied in each case. The properties involved are the zero property, identity property, associative property, and commutative property of multiplication.
- Solving Part (a) - Zero Property
For part (a), we have \(7 \times 0 = \boxed{0}\). According to the zero property of multiplication, any number multiplied by zero equals zero. Thus, the missing value is 0.
- Solving Part (b) - Identity Property
For part (b), we have \(\boxed{1} \times 8 = 8\). According to the identity property of multiplication, any number multiplied by 1 remains unchanged. Thus, the missing value is 1.
- Solving Part (c) - Associative Property
For part (c), we have \(\boxed{2} \times (3 \times 4) = (2 \times 3) \times \boxed{4}\). The associative property of multiplication states that the way factors are grouped in a multiplication expression does not change the product, i.e., \(a \times (b \times c) = (a \times b) \times c\). By comparing the sides, we see that the missing factor on the left is 2, and the missing factor on the right is 4.
- Solving Part (d) - Commutative Property
For part (d), we have \(9 \times 8 = 8 \times \boxed{9}\). The commutative property of multiplication states that the order of the factors does not change the product, i.e., \(a \times b = b \times a\). Thus, the missing value is 9.
Examples
Understanding multiplication properties is essential for simplifying algebraic expressions and performing mental math. For example, if you are calculating \(25 \times 17 \times 4\), you can use the commutative and associative properties to rearrange it as \((25 \times 4) \times 17 = 100 \times 17 = 1700\). This makes complex-looking problems much easier to solve quickly in real-world budgeting or inventory tasks.