Certainly! Let's carefully analyze the given mathematical statement and determine which property of multiplication it illustrates.
The given statement is:
[tex]$(6 \cdot y) \cdot 7 = 6 \cdot (y \cdot 7)$[/tex]
We are given four options to choose from:
A. associative property of multiplication
B. commutative property of multiplication
C. distributive property
D. identity element of multiplication
To determine the correct property, let's describe each option in the context of multiplication:
1. Associative property of multiplication: This property states that the grouping of factors does not affect the product. In other words, for any numbers [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex]:
[tex]$ (a \cdot b) \cdot c = a \cdot (b \cdot c) $[/tex]
This is exactly the rearrangement we see in the given statement.
2. Commutative property of multiplication: This property states that the order of the factors does not affect the product. In other words, for any numbers [tex]\(a\)[/tex] and [tex]\(b\)[/tex]:
[tex]$ a \cdot b = b \cdot a $[/tex]
The given statement does not change the order of multiplication; it changes only the grouping, so this property does not apply here.
3. Distributive property: This property relates multiplication and addition and can be stated as:
[tex]$ a \cdot (b + c) = (a \cdot b) + (a \cdot c) $[/tex]
The given equation does not involve addition, so the distributive property is not illustrated here.
4. Identity element of multiplication: This property states that multiplying any number by 1 leaves the number unchanged, which can be stated as:
[tex]$ a \cdot 1 = a $[/tex]
The given statement does not involve the number 1; hence this property does not apply.
Given this analysis, the correct property illustrated by the statement:
[tex]$(6 \cdot y) \cdot 7 = 6 \cdot (y \cdot 7)$[/tex]
is the associative property of multiplication.
Therefore, the correct answer is:
A. associative property of multiplication
