Which of the following correctly demonstrates the Associative Property of Addition?(1 point)

Question

Responses

m⋅(x⋅y)=(m⋅x)⋅y
m times left parenthesis x times y right parenthesis equals left parenthesis m times x right parenthesis times y

x⋅(y+z)=(x⋅y)+z
x times left parenthesis y plus z right parenthesis equals left parenthesis x times y right parenthesis plus z

y−(x+1)=(y−x)+1
y minus left parenthesis x plus 1 right parenthesis equals left parenthesis y minus x right parenthesis plus 1

(x+y)+z+r=x+(y+z)+r
left parenthesis x plus y right parenthesis plus z plus r equals x plus left parenthesis y plus z right parenthesis plus r

Answer 1

The Associative Property of Addition states that when adding three or more numbers, the way in which the numbers are grouped does not affect the sum.

From the options you've provided, the correct demonstration of the Associative Property of Addition is:

\[ (x+y)+z+r = x+(y+z)+r \]

This shows that regardless of how \(x\), \(y\), and \(z\) are grouped, the sum remains the same.

Answer 2

Which of the following demonstrates the Associative Property of Multiplication?(1 point)
Responses

(3⋅5)⋅2=3⋅(5⋅2)
left parenthesis 3 times 5 right parenthesis times 2 equals 3 times left parenthesis 5 times 2 right parenthesis

(3⋅5)+2=(3+5)⋅2
left parenthesis 3 times 5 right parenthesis plus 2 equals left parenthesis 3 plus 5 right parenthesis times 2

(3+5)⋅2=3+(5⋅2)
left parenthesis 3 plus 5 right parenthesis times 2 equals 3 plus left parenthesis 5 times 2 right parenthesis

(3⋅5)⋅2=(3⋅5)⋅2
left parenthesis 3 times 5 right parenthesis times 2 equals left parenthesis 3 times 5 right parenthesis times 2

Answer 3

The Associative Property of Multiplication states that when multiplying three or more numbers, the way in which the numbers are grouped does not affect the product.

From the options you've provided, the correct demonstration of the Associative Property of Multiplication is:

\[ (3 \cdot 5) \cdot 2 = 3 \cdot (5 \cdot 2) \]

This shows that the way in which the numbers are grouped does not change the result of the multiplication.

Answer 4

Use the Associative Property to determine which expressions are equivalent.(1 point)
Responses

−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=(−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=(−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 left parenthesis negative 2 times negative 6 y right parenthesis

−3⋅(4x⋅−2)⋅−6y=−7xy

Answer 5

To determine which expressions are equivalent using the Associative Property, we need to examine the options provided. The Associative Property allows us to change the grouping of numbers when multiplying, without changing the result.

Let’s analyze each option:

1. −3⋅(4x⋅−2)⋅−6y = 18 − 8x

   • This is an equation comparing two expressions. Without further simplification, we cannot assert that this is equivalent.

2. −3⋅(4x⋅−2)⋅−6y = (−3⋅4x)⋅−2 − 6y

   • Here, the left side is still grouping in a way that isn't appropriate for associative property (also, this seems to combine multiplication and addition incorrectly).

3. −3⋅(4x⋅−2)⋅−6y = (−3⋅4x)(−2⋅−6y)

   • This breaks it down into two groupings correctly for using multiplication. It shows how one side can be rearranged into a product of other products, which aligns with the Associative Property.

4. −3⋅(4x⋅−2)⋅−6y = −7xy

   • Similar to the first option, this compares two expressions but doesn't involve the associative property directly.

From this analysis, the expression that most clearly demonstrates the application of the Associative Property is:

−3⋅(4x⋅−2)⋅−6y = (−3⋅4x)(−2⋅−6y)

This option correctly represents the application of the Associative Property of Multiplication.

Answer 6

According to the Associative Property, which expression is equivalent to 30m+(21m−53)+(18−2m) ?(1 point)
Responses

A. 53m−35

B. (49m−53)+16

C. (30m+21m)+(18−2m)

D. 51m+(−53+18)−2m

Answer 7

To determine which expression is equivalent to \( 30m + (21m - 53) + (18 - 2m) \) according to the Associative Property, we need to see how we can group the terms differently without changing their overall value.

Using the Associative Property, we can regroup the terms. The original expression can be rearranged as follows:

1. Option A: \( 53m - 35 \)

   • This is a numerical evaluation rather than a demonstration of the associative property.

2. Option B: \( (49m - 53) + 16 \)

   • This combines the \( m \) terms to get \( 49m \) directly and adjusts the constants, but it doesn't demonstrate grouping.

3. Option C: \( (30m + 21m) + (18 - 2m) \)

   • This expression correctly uses the Associative Property by grouping the \( m \)-terms together and keeps the other terms separate.

4. Option D: \( 51m + (-53 + 18) - 2m \)

   • This expression also rearranges terms, but it does not clearly show the application of the Associative Property.

Since we are looking for the option that employs the Associative Property correctly (by creating groups without changing the overall expression), the correct choice is:

C. \( (30m + 21m) + (18 - 2m) \).

This shows how the terms can be regrouped without changing their values.