Question

Use the Associative Property to determine which expressions are equivalent.(1 point) Responses −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=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=(−3⋅4x)⋅−2−6y

Answer 1

The Associative Property allows us to regroup the terms in a multiplication without changing the result. The associative property states that for three or more numbers, the way in which they are grouped does not change their product.

Let's analyze the expressions given:

1. \(-3 \cdot (4x \cdot -2) \cdot -6y\)
2. \(-3 \cdot (4x \cdot -2) \cdot -6y\) = \(18 - 8x\)
3. \(-3 \cdot (4x \cdot -2) \cdot -6y\) = \((-3 \cdot 4x)(-2 \cdot -6y)\)
4. \(-3 \cdot (4x \cdot -2) \cdot -6y\) = \((-3 \cdot 4x) \cdot -2 \cdot -6y\)

Now let's determine if the expressions are equivalent.

Expression 1:

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

Calculating this step-by-step:

1. First compute \(4x \cdot -2 = -8x\).
2. Then multiply this by \(-3\): \(-3 \cdot -8x = 24x\).
3. Finally, multiply by \(-6y\): \(24x \cdot -6y = -144xy\).

Expression 2:

This expression states: \(-3 \cdot (4x \cdot -2) \cdot -6y = 18 - 8x\)

From our first calculation, we found that \(-3 \cdot (4x \cdot -2) \cdot -6y\) simplifies to \(-144xy\), which is not equal to \(18 - 8x\). Therefore, these are not equivalent.

Expression 3:

This expression states: \(-3 \cdot (4x \cdot -2) \cdot -6y = (-3 \cdot 4x)(-2 \cdot -6y)\)

From the Associative Property, we can re-group:

1. Compute \(-3 \cdot 4x = -12x\)
2. Compute \(-2 \cdot -6y = 12y\)
3. Now, multiply: \((-12x)(12y) = -144xy\).

This matches our calculation from Expression 1, therefore these are equivalent.

Expression 4:

This expression states: \(-3 \cdot (4x \cdot -2) \cdot -6y = (-3 \cdot 4x) \cdot -2 \cdot -6y\)

Using Associative Property:

1. Compute \(-3 \cdot 4x = -12x\)
2. Now calculate \(-12x \cdot -2 = 24x\)
3. Finally, multiply by \(-6y\): \(24x \cdot -6y = -144xy\).

This also matches our initial computation from Expression 1, therefore these are equivalent.

Summary:

Thus, the equivalent expressions based on the calculations are:

• Expression 1 \(-3 \cdot (4x \cdot -2) \cdot -6y\) is equivalent to Expression 3 and Expression 4.
• Expression 2 is not equivalent.

So, the correct responses indicating equivalence are:

• \( \text{Expressions 1, 3, and 4 are equivalent.} \)