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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 left parenthesis 3 times 5 right parenthesis times 2

(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)

5 answers

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

Among the given options, the statement that demonstrates the Associative Property of Multiplication is:

(3⋅5)⋅2 = 3⋅(5⋅2)

This shows that regardless of how the numbers are grouped, the result remains the same, thus demonstrating the associative property for multiplication.

Use the Associative Property to determine which expressions are equivalent.(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=(−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

To determine which expressions are equivalent using the Associative Property, we need to analyze each option.

  1. −3⋅(4x⋅−2)⋅−6y = (−3⋅4x)⋅−2−6y

    This expression is incorrect because there's an algebraic mistake in forming the right-hand side. The Associative Property does not change addition or multiplication in this manner.

  2. −3⋅(4x⋅−2)⋅−6y = 18−8x

    This expression appears to incorrectly simplify the left-hand side, leading to a result that is not equivalent.

  3. −3⋅(4x⋅−2)⋅−6y = (−3⋅4x)(−2⋅−6y)

    This expression correctly demonstrates the Associative Property, since it reorganizes the grouping of multiplication. We can group the multiplication in different ways without changing the overall product.

  4. −3⋅(4x⋅−2)⋅−6y = −7xy

    This expression is also not equivalent, as the left-hand side does not simplify to −7xy.

From this analysis, the correct statement that fully utilizes the Associative Property and maintains equivalence is:

−3⋅(4x⋅−2)⋅−6y = (−3⋅4x)(−2⋅−6y).

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

(49m−53)+16
left parenthesis 49 m minus 53 right parenthesis plus 16

(30m+21m)+(18−2m)
left parenthesis 30 m plus 21 m right parenthesis plus left parenthesis 18 minus 2 m right parenthesis

51m+(−53+18)−2m
51 m plus left parenthesis negative 53 plus 18 right parenthesis minus 2 m

53m−35

To determine which expression is equivalent to \( 30m + (21m - 53) + (18 - 2m) \) using the Associative Property, we should first look to simplify the original expression:

  1. First, combine similar terms inside the parentheses: \[ 30m + 21m - 53 + 18 - 2m \]

  2. Next, combine the \(m\) terms: \[ (30m + 21m - 2m) - 53 + 18 = (30 + 21 - 2)m - 53 + 18 \] \[ = 49m - 53 + 18 \] \[ = 49m - 35 \]

Now, let's review the given responses to see which matches \( 49m - 35 \):

  1. (49m − 53) + 16

    This simplifies to \( 49m - 53 + 16 = 49m - 37 \). Not equivalent.

  2. (30m + 21m) + (18 − 2m)

    This retains the original structure; it doesn't simplify to a final equivalent expression in this step. However, this expression is equivalent to the original but not in a final simplified form.

  3. 51m + (−53 + 18) − 2m

    On simplifying \( -53 + 18 \): \[ 51m - 2m - 35 = 49m - 35 \] So this is equivalent.

  4. 53m − 35

    This simplifies to \(53m - 35\), which is not equivalent because the coefficient of \(m\) is incorrect.

Thus, the expression equivalent to \(30m + (21m - 53) + (18 - 2m)\) according to the Associative Property is:

51m + (−53 + 18) − 2m.