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

Question

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 left parenthesis negative 2 times negative 6 y right parenthesis

Answer 1

To determine which expressions are equivalent using the Associative Property, let's first evaluate the left side of the first expression:

\(-3 \cdot (4x \cdot -2) \cdot -6y\).

Calculating this step-by-step:

1. First, calculate \(4x \cdot -2 = -8x\).
2. Now substitute back: \(-3 \cdot (-8x) \cdot -6y\).
3. Next, calculate \(-3 \cdot -8x = 24x\).
4. Finally, calculate \(24x \cdot -6y = -144xy\).

Thus, the left side simplifies to \(-144xy\).

Now let's evaluate the right side of each expression to see which ones are equivalent:

1. Expression 1: \((−3⋅4x)⋅−2−6y\) does not correctly represent the multiplication of \(-144xy\).
2. Expression 2: \(18−8x\) does not represent \(-144xy\).
3. Expression 3: \(-7xy\) does not match \(-144xy\).
4. Expression 4: \((−3⋅4x)(−2⋅−6y)\) involves modifying the grouping. It simplifies as follows:
   • Calculate \(-2 \cdot -6y = 12y\).
   • Substitute back: \((-3 \cdot 4x) \cdot 12y\).
   • Evaluate \(-3 \cdot 4x = -12x\).
   • Finally, calculate \(-12x \cdot 12y = -144xy\).

Since this final calculation matches our left-hand side calculation, expression 4 is equivalent to the original expression.

Therefore, the equivalent expression is: \[ −3⋅(4x⋅−2)⋅−6y=(−3⋅4x)(−2⋅−6y) \]

Answer 2

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

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

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

53m−35
53 m minus 35

(49m−53)+16

Answer 3

To determine which expression is equivalent to \(30m + (21m - 53) + (18 - 2m)\) according to the Associative Property, we need to group the terms properly and rearrange them.

1. Original Expression:
   \[ 30m + (21m - 53) + (18 - 2m) \]

2. Evaluating Step-by-Step:

   • First, combine like terms.
   • Rearranging does not change the value of the expression due to the Associative Property.

Now let’s see how each option simplifies:

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Option 1:

Response: \(51m + (-53 + 18) - 2m\)

• Simplifying:
  • Combine the constants: \(-53 + 18 = -35\)
  • Expression becomes: \(51m - 35 - 2m\)
  • Combine like terms: \(51m - 2m = 49m\)

This simplifies to \(49m - 35\), which is not equivalent to the original expression.

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Option 2:

Response: \((30m + 21m) + (18 - 2m)\)

• Simplifying:
  • Combine like terms: \(30m + 21m = 51m\)
  • Expression becomes: \(51m + (18 - 2m)\)
  • This is equivalent to the original expression as it maintains the same structure.

Thus, this option preserves equivalence.

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Option 3:

Response: \(53m - 35\)

• This results from previous groupings and simplifications but it does not show the steps of associativity.
• Since we've acknowledged in option 1 that it simplifies to \(49m - 35\), we conclude that this is not equivalent.

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Option 4:

Response: \((49m - 53) + 16\)

• This does not represent any rearrangement from the original expression directly.
• You can further check if \(49m - 53 + 16\) equals the original.
• When simplified, this becomes \(49m - 37\), which does not equal the original expression.

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Conclusion:

The expression that remains equivalent purely due to the Associative Property is:

\[ (30m + 21m) + (18 - 2m) \]

So the correct choice is:

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