Question

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 left parenthesis negative 2 times negative 6 y right parenthesis −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

Answer 1

To determine which expressions are equivalent using the Associative Property, we can analyze the expressions provided.

The Associative Property states that the way numbers are grouped in multiplication does not affect the product. This means that for any real numbers \( a, b, \) and \( c \), the following holds true:

\[
a \cdot (b \cdot c) = (a \cdot b) \cdot c
\]

Using this property, let's evaluate the first expression you provided:

1. **Expression 1:**
\[
-3 \cdot (4x \cdot -2) \cdot -6y
\]

First, we can calculate \( 4x \cdot -2 = -8x \):
\[
-3 \cdot (-8x) \cdot -6y
\]
Next, multiply:
\[
-3 \cdot -8x = 24x
\]
Finally multiply by \(-6y\):
\[
24x \cdot -6y = -144xy
\]

2. **Expression 2:**
\[
(-3 \cdot 4x) \cdot (-2 \cdot -6y)
\]
Calculate each part:
\[
-3 \cdot 4x = -12x \quad \text{and} \quad -2 \cdot -6y = 12y
\]
Now multiply them:
\[
-12x \cdot 12y = -144xy
\]

3. **Expression 3:**
\[
(-3 \cdot 4x) \cdot -2 \cdot -6y
\]
This is similar to the previous expression:
First, calculate:
\(-3 \cdot 4x = -12x\),
then:
\(-12x \cdot -2 = 24x\),
finally:
\(24x \cdot -6y = -144xy\).

4. **Expressions 4 and 5:**
You proposed numbers like \( 18 - 8x \) and \( -7xy \), but further simplification would take specific values and do not align with the previous results of \( -144xy \).

After evaluating the above expressions using the Associative Property, we find that expressions 1, 2, and 3 all yield \(-144xy\), which means they are equivalent.

So, the expressions that are equivalent are:

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

Thus the correct pairs are:
1. **Expression 1 = Expression 2**
2. **Expression 1 = Expression 3**
3. **Expression 2 = Expression 3**

Expressions involving constants like \( 18 - 8x \) or \( -7xy \) do not yield equivalent results due to the numerical values diverging from \(-144xy\).