Question

Use the Associative Property to determine which expressions are equivalent.(1 point)%0D%0AResponses%0D%0A%0D%0A−3⋅(4x⋅−2)⋅−6y=18−8x%0D%0Anegative 3 times left parenthesis 4 x times negative 2 right parenthesis times negative 6 y equals 18 minus 8 x%0D%0A%0D%0A−3⋅(4x⋅−2)⋅−6y=−7xy%0D%0Anegative 3 times left parenthesis 4 x times negative 2 right parenthesis times negative 6 y equals negative 7 x y%0D%0A%0D%0A−3⋅(4x⋅−2)⋅−6y=(−3⋅4x)(−2⋅−6y)%0D%0Anegative 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%0D%0A%0D%0A−3⋅(4x⋅−2)⋅−6y=(−3⋅4x)⋅−2−6y

Answer 1

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

1. Expression 1:

\[ -3 \cdot (4x \cdot -2) \cdot -6y = 18 - 8x \]

2. Expression 2:

\[ -3 \cdot (4x \cdot -2) \cdot -6y = -7xy \]

3. Expression 3:

\[ -3 \cdot (4x \cdot -2) \cdot -6y = (-3 \cdot 4x)(-2 \cdot -6y) \]

4. Expression 4:

\[ -3 \cdot (4x \cdot -2) \cdot -6y = (-3 \cdot 4x) \cdot -2 \cdot -6y \]

Analysis of Expressions:

Expression 1 vs. 2:

To check if the left-hand side equals the right-hand side, we need to perform the calculations.

1. Calculate the left-hand side: \(-3 \cdot (4x \cdot -2) \cdot -6y\)

   • Calculate \(4x \cdot -2 = -8x\)
   • Now, calculate \(-3 \cdot -8x \cdot -6y = -3 \cdot -8 \cdot -6 \cdot xy = 144xy\) (disregarding signs for a moment).

Now consider the right-hand sides:

• For \(18 - 8x\), we can't equate this with \(144xy\).
• For \(-7xy\), again, we don't get a match with \(144xy\).

Expression 3:

The left-hand side and right-hand side expr (-3 \cdot (4x \cdot -2) \cdot -6y) become \((-3 \cdot 4x)(-2 \cdot -6y)\) respectively.

• Calculate (\(-3 \cdot 4x) = -12x)\)
• Now calculate \(-2 \cdot -6y = 12y\)
• So we have \((-12x)(12y) = -144xy\).

Therefore, this expression does equate well.

Expression 4:

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

• First, \(-3 \cdot 4x = -12x\)
• Then \(-12x \cdot -2 = 24x\)
• Finally, \(24x \cdot -6y = -144xy\).

Both Expression 3 and Expression 4 yield \(-144xy\) on the left.

Conclusion:

Based on this analysis using the Associative Property, Expressions 3 and 4 are equivalent because they result in the same calculation, whereas Expressions 1 and 2 are not equivalent.

Answer 2

write the correct expression

Answer 3

To write the correct expression based on the calculations and comparisons we've done, we are primarily focusing on ensuring that we find an equivalent expression based on the left-hand side.

The original expression we started with is: \[ -3 \cdot (4x \cdot -2) \cdot -6y \]

From our calculations, we identified two expressions that are equivalent to the left-hand side, which results in \(-144xy\):

1. Using the Associative Property: \[ (-3 \cdot 4x)(-2 \cdot -6y) = (-12x)(12y) = -144xy \]

2. Also: \[ (-3 \cdot 4x) \cdot -2 \cdot -6y = (-12x) \cdot -2 \cdot -6y = -144xy \]

So, we can express the equivalence clearly:

The correct equivalent expression can be stated as: \[ -3 \cdot (4x \cdot -2) \cdot -6y = (-3 \cdot 4x)(-2 \cdot -6y) = (-3 \cdot 4x) \cdot -2 \cdot -6y = -144xy \]

If you want it in an algebraic form, you could simply state: \[ -3 \cdot (4x \cdot -2) \cdot -6y = -144xy \]

This conveys the equivalence concisely.