Use the Associative Property to determine which expression is correctly simplified.(1 point)
Responses
−3⋅(4x⋅−2)⋅−6y=(−3⋅4x)⋅−2−6y
negative 3 times left parenthesis 4 x times negative 2 right parenthesis times negative 6 y equals left parenthesis negative 3 times 4 x right parenthesis times negative 2 minus 6 y
−3⋅(4x⋅−2)⋅−6y=18−8x
negative 3 times left parenthesis 4 x times negative 2 right parenthesis times negative 6 y equals 18 minus 8 x
−3⋅(4x⋅−2)⋅−6y=−7xy
negative 3 times left parenthesis 4 x times negative 2 right parenthesis times negative 6 y equals negative 7 x y
−3⋅(4x⋅−2)⋅−6y=(−3⋅4x)(−2⋅−6y)
negative 3 times left parenthesis 4 x times negative 2 right parenthesis times negative 6 y equals left parenthesis negative 3 times 4 x right parenthesis times left parenthesis negative 2 times negative 6 y right parenthesis
To determine which expression is correctly simplified using the Associative Property, we need to rearrange the grouping of the terms without changing their values.
Let's look at each option:
1. −3⋅(4x⋅−2)⋅−6y = (−3⋅4x)⋅−2−6y
This expression does not correctly use the Associative Property because it rearranges the grouping of terms that aren't originally grouped together.
2. −3⋅(4x⋅−2)⋅−6y = 18−8x
This expression simplifies the terms according to the correct grouping, so it correctly uses the Associative Property.
3. −3⋅(4x⋅−2)⋅−6y = −7xy
This expression does not correctly simplify the terms according to the Associative Property. The grouping of terms is incorrect.
4. −3⋅(4x⋅−2)⋅−6y = (−3⋅4x)(−2⋅−6y)
This expression correctly uses the Associative Property by rearranging the grouping of terms.
Therefore, the correct expression that uses the Associative Property to simplify is:
−3⋅(4x⋅−2)⋅−6y = (−3⋅4x)(−2⋅−6y)
To determine which expression is correctly simplified using the Associative Property, we need to understand how the Associative Property works.
The Associative Property states that the grouping of numbers or variables does not affect the outcome of multiplication or addition. In this case, we are looking at multiplication.
Let's analyze each option:
1. −3⋅(4x⋅−2)⋅−6y = (−3⋅4x)⋅−2−6y
Here, the expression on the left is multiplying three terms together: −3, (4x⋅−2), and −6y. The expression on the right is grouping (−3⋅4x) together and then multiplying it by −2, and then subtracting 6y. This does not properly apply the Associative Property, as the grouping should not be changed. Therefore, this option is incorrect.
2. −3⋅(4x⋅−2)⋅−6y = 18−8x
In this option, the expression on the left is multiplying three terms together: −3, (4x⋅−2), and −6y. The expression on the right is simplified to a single term: 18−8x. This is a correct simplification using the Associative Property. Therefore, this option is correct.
3. −3⋅(4x⋅−2)⋅−6y = −7xy
Here, the expression on the left is multiplying three terms together: −3, (4x⋅−2), and −6y. The expression on the right, −7xy, is a completely different expression. This simplification does not follow the Associative Property. Therefore, this option is incorrect.
4. −3⋅(4x⋅−2)⋅−6y = (−3⋅4x)(−2⋅−6y)
In this option, the expression on the left is multiplying three terms together: −3, (4x⋅−2), and −6y. The expression on the right is using the Distributive Property to simplify further by multiplying (−3⋅4x) with (−2⋅−6y). This does not correctly apply the Associative Property, as the grouping of terms has changed. Therefore, this option is incorrect.
Based on our analysis, the correct option is:
−3⋅(4x⋅−2)⋅−6y = 18−8x.