Let's solve the problem step-by-step to determine which algebraic property is being used.
The given equation is:
[tex]\[ 4 \cdot (5 \cdot 2) = (4 \cdot 5) \cdot 2 \][/tex]
We need to identify the algebraic property this equation represents.
To do that, let's analyze the structure and description of each option:
### a. Commutative Property
The commutative property states that changing the order of the numbers in an operation does not change the result. For multiplication, it can be written as:
[tex]\[ a \cdot b = b \cdot a \][/tex]
In the given equation, the order of the numbers is not being changed; rather, the way in which the numbers are grouped is being altered. Hence, this property does not apply here.
### b. Distributive Property
The distributive property connects multiplication and addition, stating:
[tex]\[ a \cdot (b + c) = a \cdot b + a \cdot c \][/tex]
This property is not relevant to the given equation as it does not involve addition within the parentheses.
### c. Associative Property
The associative property states that the way in which numbers are grouped in multiplication or addition does not change the result. For multiplication, it can be written as:
[tex]\[ (a \cdot b) \cdot c = a \cdot (b \cdot c) \][/tex]
In the given equation:
[tex]\[ 4 \cdot (5 \cdot 2) = (4 \cdot 5) \cdot 2 \][/tex]
We see that the grouping of the numbers is changed, but the order remains the same, indicating that this is indeed an example of the associative property of multiplication.
### d. Inverse Property
The inverse property involves combining an element with its inverse to obtain the identity element. For multiplication, it looks like:
[tex]\[ a \cdot a^{-1} = 1 \][/tex]
This property is not relevant to our equation as it doesn't involve finding or using inverses.
### e. Identity Property
The identity property states that multiplying any number by the identity element (which is 1 for multiplication) gives the number itself:
[tex]\[ a \cdot 1 = a \][/tex]
This property is not relevant as the equation does not involve the number 1.
Given this analysis, the property being used in the equation:
[tex]\[ 4 \cdot (5 \cdot 2) = (4 \cdot 5) \cdot 2 \][/tex]
is the associative property of multiplication.
The correct answer is:
c. associative
