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

Question

Responses

−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 times negative 2 minus 6 y

−3⋅(4x⋅−2)⋅−6y=(−3⋅4x)(−2⋅−6y)

Answer 1

To analyze which expressions are equivalent using the Associative Property, we need to remember that the Associative Property allows us to group numbers in expressions differently without changing the outcome. The expressions can be simplified to determine their equivalence.

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

Now, let’s simplify the left side of the expressions, \(-3 \cdot (4x \cdot -2) \cdot -6y\):

1. Calculate \(4x \cdot -2\):
   \(4x \cdot -2 = -8x\)

2. Then, calculate \(-3 \cdot (-8x) \cdot -6y\):
   \(-3 \cdot -8x = 24x\)
   Then, \(24x \cdot -6y = -144xy\)

So, we find that:
\(-3 \cdot (4x \cdot -2) \cdot -6y = -144xy\)

Now, let's check the right sides of each original expression:

• For Expression 1: \(18 - 8x\) ∼ \(-144xy\) (Not equivalent)
• For Expression 2: \(-7xy\) ∼ \(-144xy\) (Not equivalent)
• For Expression 3: \((-3 \cdot 4x) \cdot -2 - 6y\)
  This simply doesn't form a single expression equal to \(-144xy\) (Not equivalent)
• For Expression 4: \((-3 \cdot 4x)(-2 \cdot -6y)\)
  Let's break this down:
  \((-3 \cdot 4x) = -12x\),
  and \(-2 \cdot -6y = 12y\), hence \((-12x)(12y) = -144xy\)

The only equivalent expression here is Expression 4: \[ -3 \cdot (4x \cdot -2) \cdot -6y = (-3 \cdot 4x)(-2 \cdot -6y) \]

So the final answer is:

• Expression 4 is equivalent.