Which of the following expressions could you rewrite using the Associative Property?(1 point)

Question

Responses

(5+8)⋅3
left parenthesis 5 plus 8 right parenthesis times 3

(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 times 8 right parenthesis plus 3

(9⋅5)+8+3
abcord

Answer 1

The Associative Property refers to the way in which numbers are grouped in addition or multiplication, without changing the result. It states that for any numbers a, b, and c:

• For addition: \( (a + b) + c = a + (b + c) \)
• For multiplication: \( (a \cdot b) \cdot c = a \cdot (b \cdot c) \)

Looking at the provided options:

1. (5 + 8) · 3: This expression cannot be rewritten using the Associative Property because it involves multiplication of a sum with another number.

2. (9 · 5) · (8 · 3): This expression could potentially use the Associative Property, but it cannot be effectively simplified by grouping the terms differently.

3. (5 · 8) + 3: This expression cannot be rewritten using the Associative Property because it involves a multiplication followed by an addition.

4. (9 · 5) + 8 + 3: This expression can be rewritten using the Associative Property of addition, as you can group the numbers differently while adding: \( (9 · 5) + (8 + 3) \) or \( ( (9 · 5) + 8 ) + 3 \).

Therefore, the expression that could be rewritten using the Associative Property is:

(9·5) + 8 + 3.

Answer 2

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

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

(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

y−(x+1)=(y−x)+1

Answer 3

The Associative Property of Addition states that the way in which numbers are grouped in addition does not affect the sum. Mathematically, it can be expressed as:

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

Now, let's analyze the options provided:

1. \( m \cdot (x \cdot y) = (m \cdot x) \cdot y \): This expression demonstrates the Associative Property of Multiplication, not addition.

2. \( x \cdot (y + z) = (x \cdot y) + z \): This expression incorrectly represents the situation; it mixes multiplication with addition and does not illustrate the Associative Property of Addition.

3. \( (x + y) + z + r = x + (y + z) + r \): This expression correctly demonstrates the Associative Property of Addition, as it shows that the way in which \( x, y, z, \) and \( r \) are grouped in addition does not change the sum.

4. \( y - (x + 1) = (y - x) + 1 \): This expression does not represent the Associative Property of Addition; it is an equation that deals with subtraction.

The correct response that demonstrates the Associative Property of Addition is:

\( (x + y) + z + r = x + (y + z) + r \).

Answer 4

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 left parenthesis 3 times 5 right parenthesis times 2

(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 5

The Associative Property of Multiplication states that the way in which numbers are grouped in multiplication does not affect the product. Mathematically, it can be represented as:

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

Now, let's analyze the options provided:

1. \((3 + 5) \cdot 2 = 3 + (5 \cdot 2)\): This expression demonstrates the Distributive Property, not the Associative Property of Multiplication.

2. \((3 \cdot 5) \cdot 2 = (3 \cdot 5) \cdot 2\): This expression is merely a repetition of the same statement and does not illustrate the Associative Property.

3. \((3 \cdot 5) + 2 = (3 + 5) \cdot 2\): This expression does not demonstrate the Associative Property of Multiplication, as it mixes addition with multiplication incorrectly.

4. \((3 \cdot 5) \cdot 2 = 3 \cdot (5 \cdot 2)\): This expression correctly demonstrates the Associative Property of Multiplication, showing that it doesn't matter how the numbers are grouped when multiplying.

The correct response that demonstrates the Associative Property of Multiplication is:

\((3 \cdot 5) \cdot 2 = 3 \cdot (5 \cdot 2)\).

Answer 6

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 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

−3⋅(4x⋅−2)⋅−6y=(−3⋅4x)(−2⋅−6y)

Answer 7

To evaluate the equivalence of the expressions using the Associative Property, we need to analyze how the grouping of terms affects the outcome. The Associative Property allows us to regroup terms in multiplication without changing the product.

Let's break down each response:

1. \(-3 \cdot (4x \cdot -2) \cdot -6y = (-3 \cdot 4x) \cdot -2 -6y\): This equation incorrectly applies the Associative Property as it looks inconsistent with proper multiplication. The right side lacks correct use of parentheses for multiplication and does not add further terms correctly.

2. \(-3 \cdot (4x \cdot -2) \cdot -6y = 18 - 8x\): To solve the left side, we calculate \(-3 \cdot 4x = -12x\) and then \(-12x \cdot -2 = 24x\), and finally \(24x \cdot -6y\) which does not equate to \(18 - 8x\).

3. \(-3 \cdot (4x \cdot -2) \cdot -6y = -7xy\): Similarly, we evaluate the left part. Following the same procedure shows that it doesn't lead to \(-7xy\).

4. \(-3 \cdot (4x \cdot -2) \cdot -6y = (-3 \cdot 4x)(-2 \cdot -6y)\): This expression can be verified using the Associative Property.

   • Left Side: \[-3 \cdot (4x \cdot -2) \cdot -6y = (-3 \cdot 4x \cdot -2) \cdot -6y\]
   • Right Side: \[(-3 \cdot 4x) \cdot (-2 \cdot -6y)\] Both sides illustrate proper use of multiplication regrouping under the Associative Property and end up resulting in equivalent expressions since the products remain unchanged.

Only the last option effectively uses the Associative Property and verifies equivalence.

Thus, the equivalent expression using the Associative Property is:

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

Answer 8

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

(49m−53)+16
left parenthesis 49 m minus 53 right parenthesis plus 16

51m+(−53+18)−2m

Answer 9

To determine which expression is equivalent to \(30m + (21m - 53) + (18 - 2m)\) using the Associative Property, we can regroup the terms in the original expression without changing the sum.

The original expression is: \[ 30m + (21m - 53) + (18 - 2m) \]

Now let's analyze the given options one by one:

1. \(53m - 35\): To assess this expression, we can compute \(30m + 21m - 2m - 53 + 18\) from the original expression. Simplifying gives \(53m - 35\). However, verifying against the Associative Property wasn't demonstrated.

2. \((30m + 21m) + (18 - 2m)\): This expression correctly demonstrates the Associative Property of Addition. It simply groups the terms \(30m\) and \(21m\) together first, followed by the subtraction \(18 - 2m\) as a subsequent term. This grouping maintains equivalence to the original expression.

3. \((49m - 53) + 16\): Let's break this down to check for equivalence. If we compute \(49m - 53 + 16\), we get \(49m - 37\), which does not match the original.

4. \(51m + (-53 + 18) - 2m\): Simplifying yields: \(51m - 2m + (-53 + 18) = 49m - 35\), which also does not match the original expression.

Based on these evaluations, the expression that correctly utilizes the Associative Property while simplifying the original expression is:

\((30m + 21m) + (18 - 2m)\).