Question
Use the Associative Property to determine which expressions are equivalent.(1 point)
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 left parenthesis negative 2 times negative 6 y right parenthesis
−3⋅(4x⋅−2)⋅−6y=(−3⋅4x)⋅−2−6y
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 left parenthesis negative 2 times negative 6 y right parenthesis
−3⋅(4x⋅−2)⋅−6y=(−3⋅4x)⋅−2−6y
Answers
Answer
According to the Associative Property, which expression is equivalent to 30m+(21m−53)+(18−2m) ?(1 point)
Responses
53m−35
53 m minus 35
(30m+21m)+(18−2m)
left parenthesis 30 m plus 21 m right parenthesis plus left parenthesis 18 minus 2 m right parenthesis
51m+(−53+18)−2m
51 m plus left parenthesis negative 53 plus 18 right parenthesis minus 2 m
(49m−53)+16
Responses
53m−35
53 m minus 35
(30m+21m)+(18−2m)
left parenthesis 30 m plus 21 m right parenthesis plus left parenthesis 18 minus 2 m right parenthesis
51m+(−53+18)−2m
51 m plus left parenthesis negative 53 plus 18 right parenthesis minus 2 m
(49m−53)+16
Answer
Which of the following demonstrates the Associative Property of Multiplication?(1 point)
Responses
(3+5)⋅2=3+(5⋅2)
left parenthesis 3 plus 5 right parenthesis times 2 equals 3 plus left parenthesis 5 times 2 right parenthesis
(3⋅5)⋅2=3⋅(5⋅2)
left parenthesis 3 times 5 right parenthesis times 2 equals 3 times left parenthesis 5 times 2 right parenthesis
(3⋅5)+2=(3+5)⋅2
left parenthesis 3 times 5 right parenthesis plus 2 equals left parenthesis 3 plus 5 right parenthesis times 2
(3⋅5)⋅2=(3⋅5)⋅2
Responses
(3+5)⋅2=3+(5⋅2)
left parenthesis 3 plus 5 right parenthesis times 2 equals 3 plus left parenthesis 5 times 2 right parenthesis
(3⋅5)⋅2=3⋅(5⋅2)
left parenthesis 3 times 5 right parenthesis times 2 equals 3 times left parenthesis 5 times 2 right parenthesis
(3⋅5)+2=(3+5)⋅2
left parenthesis 3 times 5 right parenthesis plus 2 equals left parenthesis 3 plus 5 right parenthesis times 2
(3⋅5)⋅2=(3⋅5)⋅2
Answer
Which of the following correctly demonstrates the Associative Property of Addition?(1 point)
Responses
m⋅(x⋅y)=(m⋅x)⋅y
m times left parenthesis x times y right parenthesis equals left parenthesis m times x right parenthesis times y
y−(x+1)=(y−x)+1
y minus left parenthesis x plus 1 right parenthesis equals left parenthesis y minus x right parenthesis plus 1
(x+y)+z+r=x+(y+z)+r
left parenthesis x plus y right parenthesis plus z plus r equals x plus left parenthesis y plus z right parenthesis plus r
x⋅(y+z)=(x⋅y)+z
x times left parenthesis y plus z right parenthesis equals left parenthesis x times y right parenthesis plus z
Responses
m⋅(x⋅y)=(m⋅x)⋅y
m times left parenthesis x times y right parenthesis equals left parenthesis m times x right parenthesis times y
y−(x+1)=(y−x)+1
y minus left parenthesis x plus 1 right parenthesis equals left parenthesis y minus x right parenthesis plus 1
(x+y)+z+r=x+(y+z)+r
left parenthesis x plus y right parenthesis plus z plus r equals x plus left parenthesis y plus z right parenthesis plus r
x⋅(y+z)=(x⋅y)+z
x times left parenthesis y plus z right parenthesis equals left parenthesis x times y right parenthesis plus z
Answer
Which of the following expressions could you rewrite using the Associative Property?(1 point)
Responses
(9⋅5)⋅(8⋅3)
left parenthesis 9 times 5 right parenthesis times left parenthesis 8 times 3 right parenthesis
(5+8)⋅3
left parenthesis 5 plus 8 right parenthesis times 3
(5⋅8)+3
left parenthesis 5 times 8 right parenthesis plus 3
(9⋅5)+8+
Responses
(9⋅5)⋅(8⋅3)
left parenthesis 9 times 5 right parenthesis times left parenthesis 8 times 3 right parenthesis
(5+8)⋅3
left parenthesis 5 plus 8 right parenthesis times 3
(5⋅8)+3
left parenthesis 5 times 8 right parenthesis plus 3
(9⋅5)+8+
Answered by
GPT-4o mini
The Associative Property states that when you are multiplying numbers, the way you group them does not change their product. In other words, if you have three or more factors, you can group them in any way you like without changing the result. For example, \( a \cdot (b \cdot c) = (a \cdot b) \cdot c \).
Let’s analyze the expressions provided:
1. **Expression: \( -3 \cdot (4x \cdot -2) \cdot -6y = 18 - 8x \)**
This appears to be evaluating an expression to see if it equals \( 18 - 8x \). Without further calculation, we cannot confirm if this is equivalent.
2. **Expression: \( -3 \cdot (4x \cdot -2) \cdot -6y = -7xy \)**
Similarly, this is another evaluation that is not directly related to the Associative Property but shows if the multiplication results in these terms.
3. **Expression: \( -3 \cdot (4x \cdot -2) \cdot -6y = (-3 \cdot 4x)(-2 \cdot -6y) \)**
This one applies the Associative Property correctly by regrouping the factors. Therefore, this expression is equivalent as it merely changes the grouping of the terms.
4. **Expression: \( -3 \cdot (4x \cdot -2) \cdot -6y = (-3 \cdot 4x) \cdot -2 \cdot -6y \)**
This does not maintain the logical grouping for the Associative Property and alters the way the products are evaluated.
Thus, the expression that definitely holds through the Associative Property is:
**−3⋅(4x⋅−2)⋅−6y=(−3⋅4x)(−2⋅−6y)**
This confirms that they are equivalent while applying the associative property correctly.
Let’s analyze the expressions provided:
1. **Expression: \( -3 \cdot (4x \cdot -2) \cdot -6y = 18 - 8x \)**
This appears to be evaluating an expression to see if it equals \( 18 - 8x \). Without further calculation, we cannot confirm if this is equivalent.
2. **Expression: \( -3 \cdot (4x \cdot -2) \cdot -6y = -7xy \)**
Similarly, this is another evaluation that is not directly related to the Associative Property but shows if the multiplication results in these terms.
3. **Expression: \( -3 \cdot (4x \cdot -2) \cdot -6y = (-3 \cdot 4x)(-2 \cdot -6y) \)**
This one applies the Associative Property correctly by regrouping the factors. Therefore, this expression is equivalent as it merely changes the grouping of the terms.
4. **Expression: \( -3 \cdot (4x \cdot -2) \cdot -6y = (-3 \cdot 4x) \cdot -2 \cdot -6y \)**
This does not maintain the logical grouping for the Associative Property and alters the way the products are evaluated.
Thus, the expression that definitely holds through the Associative Property is:
**−3⋅(4x⋅−2)⋅−6y=(−3⋅4x)(−2⋅−6y)**
This confirms that they are equivalent while applying the associative property correctly.
Answered by
GPT-4o mini
To determine which expression is equivalent to \( 30m + (21m - 53) + (18 - 2m) \) using the Associative Property, we need to recognize that the Associative Property allows us to group terms in any combination without changing the overall sum.
1. **Original expression:**
\[
30m + (21m - 53) + (18 - 2m)
\]
2. Now let's evaluate the provided responses one by one:
- **Response: \( 53m - 35 \)**
This response requires simplification of the original expression to confirm if they are equivalent.
- **Response: \( (30m + 21m) + (18 - 2m) \)**
This applies the Associative Property correctly by grouping \( 30m \) and \( 21m \) together. Though it is not fully simplified, it holds true in terms of regrouping.
- **Response: \( 51m + (-53 + 18) - 2m \)**
This response also applies regrouping but must be simplified: \( -53 + 18 = -35 \), giving:
\[
51m - 35 - 2m = (51m - 2m) - 35 = 49m - 35
\]
Therefore, this expression is not equivalent to the original expression.
- **Response: \( (49m - 53) + 16 \)**
This response implies simplification to see if it matches the original expression:
\[
49m - 53 + 16 = 49m - 37
\]
This is also not equivalent.
3. To summarize, the response that correctly applies the Associative Property is:
\[
(30m + 21m) + (18 - 2m)
\]
Thus, **the equivalent expression is:**
**\((30m + 21m) + (18 - 2m)\)**.
1. **Original expression:**
\[
30m + (21m - 53) + (18 - 2m)
\]
2. Now let's evaluate the provided responses one by one:
- **Response: \( 53m - 35 \)**
This response requires simplification of the original expression to confirm if they are equivalent.
- **Response: \( (30m + 21m) + (18 - 2m) \)**
This applies the Associative Property correctly by grouping \( 30m \) and \( 21m \) together. Though it is not fully simplified, it holds true in terms of regrouping.
- **Response: \( 51m + (-53 + 18) - 2m \)**
This response also applies regrouping but must be simplified: \( -53 + 18 = -35 \), giving:
\[
51m - 35 - 2m = (51m - 2m) - 35 = 49m - 35
\]
Therefore, this expression is not equivalent to the original expression.
- **Response: \( (49m - 53) + 16 \)**
This response implies simplification to see if it matches the original expression:
\[
49m - 53 + 16 = 49m - 37
\]
This is also not equivalent.
3. To summarize, the response that correctly applies the Associative Property is:
\[
(30m + 21m) + (18 - 2m)
\]
Thus, **the equivalent expression is:**
**\((30m + 21m) + (18 - 2m)\)**.
Answered by
GPT-4o mini
To demonstrate the Associative Property of Multiplication, we need to show that when multiplying three or more numbers, the way in which the numbers are grouped does not affect the product. The Associative Property states that:
\[
(a \cdot b) \cdot c = a \cdot (b \cdot c)
\]
Let's analyze the provided responses:
1. **Response:** \((3+5) \cdot 2 = 3 + (5 \cdot 2)\)
- This involves addition and multiplication but does not demonstrate the Associative Property of multiplication.
2. **Response:** \((3 \cdot 5) \cdot 2 = 3 \cdot (5 \cdot 2)\)
- This one correctly demonstrates the Associative Property of Multiplication. It shows that regardless of whether you multiply \(3 \cdot 5\) first or \(5 \cdot 2\) first, the result will be the same because both groupings involve multiplication.
3. **Response:** \((3 \cdot 5) + 2 = (3 + 5) \cdot 2\)
- This involves both addition and multiplication and does not demonstrate the Associative Property of Multiplication.
4. **Response:** \((3 \cdot 5) \cdot 2 = (3 \cdot 5) \cdot 2\)
- This is just a restatement of the same expression and doesn't illustrate the property.
Based on this analysis, the **correct answer** that demonstrates the Associative Property of Multiplication is:
**\((3 \cdot 5) \cdot 2 = 3 \cdot (5 \cdot 2)\)**.
\[
(a \cdot b) \cdot c = a \cdot (b \cdot c)
\]
Let's analyze the provided responses:
1. **Response:** \((3+5) \cdot 2 = 3 + (5 \cdot 2)\)
- This involves addition and multiplication but does not demonstrate the Associative Property of multiplication.
2. **Response:** \((3 \cdot 5) \cdot 2 = 3 \cdot (5 \cdot 2)\)
- This one correctly demonstrates the Associative Property of Multiplication. It shows that regardless of whether you multiply \(3 \cdot 5\) first or \(5 \cdot 2\) first, the result will be the same because both groupings involve multiplication.
3. **Response:** \((3 \cdot 5) + 2 = (3 + 5) \cdot 2\)
- This involves both addition and multiplication and does not demonstrate the Associative Property of Multiplication.
4. **Response:** \((3 \cdot 5) \cdot 2 = (3 \cdot 5) \cdot 2\)
- This is just a restatement of the same expression and doesn't illustrate the property.
Based on this analysis, the **correct answer** that demonstrates the Associative Property of Multiplication is:
**\((3 \cdot 5) \cdot 2 = 3 \cdot (5 \cdot 2)\)**.
Answered by
GPT-4o mini
To demonstrate the Associative Property of Addition, we need to show that when adding three or more numbers, the way in which the numbers are grouped does not affect the sum. The Associative Property of Addition states that:
\[
(a + b) + c = a + (b + c)
\]
Now, let's analyze the provided responses:
1. **Response:** \( m \cdot (x \cdot y) = (m \cdot x) \cdot y \)
- This expression demonstrates the Associative Property of Multiplication, not Addition.
2. **Response:** \( y - (x + 1) = (y - x) + 1 \)
- This involves subtraction and does not demonstrate the Associative Property of Addition.
3. **Response:** \( (x + y) + z + r = x + (y + z) + r \)
- This correctly demonstrates the Associative Property of Addition. It shows that the way we group the terms when adding does not change the sum.
4. **Response:** \( x \cdot (y + z) = (x \cdot y) + z \)
- This expression deals with multiplication and does not demonstrate the Associative Property of Addition.
Based on this analysis, the **correct answer** that demonstrates the Associative Property of Addition is:
**\((x + y) + z + r = x + (y + z) + r\)**.
\[
(a + b) + c = a + (b + c)
\]
Now, let's analyze the provided responses:
1. **Response:** \( m \cdot (x \cdot y) = (m \cdot x) \cdot y \)
- This expression demonstrates the Associative Property of Multiplication, not Addition.
2. **Response:** \( y - (x + 1) = (y - x) + 1 \)
- This involves subtraction and does not demonstrate the Associative Property of Addition.
3. **Response:** \( (x + y) + z + r = x + (y + z) + r \)
- This correctly demonstrates the Associative Property of Addition. It shows that the way we group the terms when adding does not change the sum.
4. **Response:** \( x \cdot (y + z) = (x \cdot y) + z \)
- This expression deals with multiplication and does not demonstrate the Associative Property of Addition.
Based on this analysis, the **correct answer** that demonstrates the Associative Property of Addition is:
**\((x + y) + z + r = x + (y + z) + r\)**.
Answered by
GPT-4o mini
The Associative Property allows you to regroup numbers when performing addition or multiplication. It states that the way in which numbers are grouped does not affect their sum or product.
1. **Response:** \((9 \cdot 5) \cdot (8 \cdot 3)\)
- This expression contains groups of multiplication, but it wouldn't be appropriate to apply the Associative Property, as it’s already grouped. The Associative Property applies to changing the grouping, for example: \((9 \cdot 5) \cdot 3 = 9 \cdot (5 \cdot 3)\). However, the expression looks a bit more complex since it includes two separate products. Thus, it can't be directly rewritten using the Associative Property.
2. **Response:** \((5 + 8) \cdot 3\)
- This expression deals with addition and multiplication but would not be written differently using the Associative Property, as the Associative Property pertains specifically to the grouping of terms of addition or multiplication, not a mix of both.
3. **Response:** \((5 \cdot 8) + 3\)
- This expression involves multiplication followed by addition, but similar to the previous case, the Associative Property does not apply here as it does not allow us to rearrange the groups involving both addition and multiplication.
4. **Response:** \((9 \cdot 5) + 8\)
- This expression contains addition and multiplication. However, it cannot be rewritten using the Associative Property directly since it involves mixed operations.
The expression that most correctly applies the Associative Property would be one that only involves either addition or multiplication.
**None of the given expressions can strictly be rewritten using the Associative Property.**
For example, an expression like \((a + b) + c = a + (b + c)\) or \((x \cdot y) \cdot z = x \cdot (y \cdot z)\) would be more fitting if present among the options.
If you had to pick one among those listed that could hypothetically relate to the Associative Property with adjusted context or interpretation, you would lean more towards those consisting of solely addition or solely multiplication without complication, like the first response dealing with the multiplication of numbers, but ultimately, no expression is shown directly applicable here.
1. **Response:** \((9 \cdot 5) \cdot (8 \cdot 3)\)
- This expression contains groups of multiplication, but it wouldn't be appropriate to apply the Associative Property, as it’s already grouped. The Associative Property applies to changing the grouping, for example: \((9 \cdot 5) \cdot 3 = 9 \cdot (5 \cdot 3)\). However, the expression looks a bit more complex since it includes two separate products. Thus, it can't be directly rewritten using the Associative Property.
2. **Response:** \((5 + 8) \cdot 3\)
- This expression deals with addition and multiplication but would not be written differently using the Associative Property, as the Associative Property pertains specifically to the grouping of terms of addition or multiplication, not a mix of both.
3. **Response:** \((5 \cdot 8) + 3\)
- This expression involves multiplication followed by addition, but similar to the previous case, the Associative Property does not apply here as it does not allow us to rearrange the groups involving both addition and multiplication.
4. **Response:** \((9 \cdot 5) + 8\)
- This expression contains addition and multiplication. However, it cannot be rewritten using the Associative Property directly since it involves mixed operations.
The expression that most correctly applies the Associative Property would be one that only involves either addition or multiplication.
**None of the given expressions can strictly be rewritten using the Associative Property.**
For example, an expression like \((a + b) + c = a + (b + c)\) or \((x \cdot y) \cdot z = x \cdot (y \cdot z)\) would be more fitting if present among the options.
If you had to pick one among those listed that could hypothetically relate to the Associative Property with adjusted context or interpretation, you would lean more towards those consisting of solely addition or solely multiplication without complication, like the first response dealing with the multiplication of numbers, but ultimately, no expression is shown directly applicable here.