Question
Use the Associative Property to determine which expressions are equivalent.
A. -3*(4x*-2)*-6y=18-8x
B. -3*(4x*-2)*-6y=-7xy
C. -3*(4x*-2)*-6y=(-3*4x)*-2-6y
D. -3*(4x*-2)*-6y=(-3*4x)(-2*-6y)
A. -3*(4x*-2)*-6y=18-8x
B. -3*(4x*-2)*-6y=-7xy
C. -3*(4x*-2)*-6y=(-3*4x)*-2-6y
D. -3*(4x*-2)*-6y=(-3*4x)(-2*-6y)
Answers
Answered by
GPT-4o mini
The Associative Property states that the way numbers are grouped in multiplication does not change the product. In other words, for any numbers \( a, b, c \):
\[
a \times (b \times c) = (a \times b) \times c
\]
Let's evaluate each of the provided expressions using the Associative Property.
### Expression A:
\[
-3*(4x*-2)*-6y=18-8x
\]
This expression equates two different types of expressions (one being a product and the other a difference), so we will not consider this expression as equivalent based on the Associative Property since they are not even both products.
### Expression B:
\[
-3*(4x*-2)*-6y=-7xy
\]
Similar to A, this is again two different types of expressions (one a product and the other a single term). So it won't be equivalent through the Associative Property.
### Expression C:
\[
-3*(4x*-2)*-6y=(-3*4x)*-2*-6y
\]
Here, the left side can be regrouped using the Associative Property, but then it needs further simplification. To assess equivalence, let's evaluate the product on the left side:
1. Calculate \( 4x \times -2 = -8x \).
2. Now, the left becomes \( -3 \times (-8x) \times -6y = 24xy \).
3. The right can also be simplified: \( (-3 \times 4x) = -12x \), and \( -2 \times -6y = 12y \).
4. Now evaluate this right side: \((-12x)(12y) = -144xy\).
Since \( 24xy \neq -144xy \), this expression does not hold equivalence.
### Expression D:
\[
-3*(4x*-2)*-6y=(-3*4x)(-2*-6y)
\]
Let's evaluate both sides:
- The left side simplifies as before:
\[
= -3 \times (-8x) \times -6y = 24xy
\]
- The right side:
1. Calculate \(-3 \times 4x = -12x\).
2. Next: \(-2 \times -6y = 12y\).
3. Evaluate \( (-12x)(12y) = -144xy \).
Again, since \( 24xy \neq -144xy \), this expression does not hold equivalence.
In conclusion, based on the Associative Property of multiplication, none of the expressions A, B, C, or D are equivalent.
\[
a \times (b \times c) = (a \times b) \times c
\]
Let's evaluate each of the provided expressions using the Associative Property.
### Expression A:
\[
-3*(4x*-2)*-6y=18-8x
\]
This expression equates two different types of expressions (one being a product and the other a difference), so we will not consider this expression as equivalent based on the Associative Property since they are not even both products.
### Expression B:
\[
-3*(4x*-2)*-6y=-7xy
\]
Similar to A, this is again two different types of expressions (one a product and the other a single term). So it won't be equivalent through the Associative Property.
### Expression C:
\[
-3*(4x*-2)*-6y=(-3*4x)*-2*-6y
\]
Here, the left side can be regrouped using the Associative Property, but then it needs further simplification. To assess equivalence, let's evaluate the product on the left side:
1. Calculate \( 4x \times -2 = -8x \).
2. Now, the left becomes \( -3 \times (-8x) \times -6y = 24xy \).
3. The right can also be simplified: \( (-3 \times 4x) = -12x \), and \( -2 \times -6y = 12y \).
4. Now evaluate this right side: \((-12x)(12y) = -144xy\).
Since \( 24xy \neq -144xy \), this expression does not hold equivalence.
### Expression D:
\[
-3*(4x*-2)*-6y=(-3*4x)(-2*-6y)
\]
Let's evaluate both sides:
- The left side simplifies as before:
\[
= -3 \times (-8x) \times -6y = 24xy
\]
- The right side:
1. Calculate \(-3 \times 4x = -12x\).
2. Next: \(-2 \times -6y = 12y\).
3. Evaluate \( (-12x)(12y) = -144xy \).
Again, since \( 24xy \neq -144xy \), this expression does not hold equivalence.
In conclusion, based on the Associative Property of multiplication, none of the expressions A, B, C, or D are equivalent.
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