Use the Associative Property to simplify the expression 3x+(x+2)

Question

to its simplest equivalent form.

Answer 1

To simplify the expression \( 3x + (x + 2) \) using the Associative Property, we will first rearrange and group the terms.

The Associative Property states that the way in which numbers are grouped in addition does not change the sum. This means we can regroup the terms however we like.

Starting with the original expression:

\[ 3x + (x + 2) \]

We can remove the parentheses and regroup the terms:

\[ 3x + x + 2 \]

Now, we can combine the like terms \(3x\) and \(x\):

\[ (3x + x) + 2 = 4x + 2 \]

Thus, the simplest equivalent form of the expression \(3x + (x + 2)\) is:

\[ \boxed{4x + 2} \]

Answer 2

Use the Associative Property to group like terms together for the expression 9x+(2x+3)
Do not simplify the grouped terms.

Answer 3

To use the Associative Property to group like terms in the expression \(9x + (2x + 3)\), we'll focus on rearranging and grouping the terms without simplifying them further.

Starting with the expression:

\[ 9x + (2x + 3) \]

According to the Associative Property, we can change the grouping of the terms. We can group \(9x\) and \(2x\) together, while keeping \(3\) as a separate term:

\[ (9x + 2x) + 3 \]

This groups the like terms \(9x\) and \(2x\) together while keeping \(3\) outside of that grouping.

Thus, the expression \(9x + (2x + 3)\) can be grouped as:

\[ \boxed{(9x + 2x) + 3} \]

Answer 4

Which of the following expressions could you rewrite using the Associative Property?(1 point)
Responses:
(5+8)⋅3
(9⋅5)+8+3
(9⋅5)⋅(8⋅3)
(5⋅8)+3

Answer 5

The Associative Property refers to the ability to regroup terms in addition or multiplication without changing the result. This can be applied to both addition and multiplication.

Let's analyze the given options:

1. \((5 + 8) \cdot 3\): This expression is in the form of addition being used within a multiplication. You could technically rewrite how the addition is grouped (e.g., with different pairings), but it's primarily a multiplication expression, so it's not the best choice.

2. \((9 \cdot 5) + 8 + 3\): This expression uses both multiplication and addition. We could apply the Associative Property to the addition (grouping \(8 + 3\) together), so this expression can be rewritten.

3. \((9 \cdot 5) \cdot (8 \cdot 3)\): This involves multiplication only, and while you can regroup the multiplications (e.g., \(9 \cdot (5 \cdot 8) \cdot 3\)), the overall multiplication does not inherently involve changing the addition groupings.

4. \((5 \cdot 8) + 3\): Similar to the first option, this expression primarily involves addition and multiplication. While you can rearrange the multiplication itself, it isn't a classic example of utilizing the Associative Property for addition.

Based on these analyses, the best expression that could be rewritten using the Associative Property is:

\[ \boxed{(9 \cdot 5) + 8 + 3} \]

This allows us to regroup the addition terms as needed.

Answer 6

Which of the following correctly demonstrates the Associative Property of Addition?(1 point)
Responses:
y−(x+1)=(y−x)+1
m⋅(x⋅y)=(m⋅x)⋅y
x⋅(y+z)=(x⋅y)+z
(x+y)+z+r=x+(y+z)+r

Answer 7

The Associative Property of Addition states that the way in which numbers are grouped in addition does not change the sum. In mathematical terms, for any numbers \(a\), \(b\), and \(c\):

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

Let’s analyze the given options:

1. \(y - (x + 1) = (y - x) + 1\): This expression involves subtraction, not addition. Therefore, it does not demonstrate the Associative Property of Addition.

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

3. \(x \cdot (y + z) = (x \cdot y) + z\): This expression incorrectly mixes operations (it uses multiplication and addition) and thus does not represent the Associative Property of Addition.

4. \((x + y) + z + r = x + (y + z) + r\): This correctly shows how the grouping of the numbers \(x\), \(y\), \(z\), and \(r\) can be changed when adding, without changing the result.

Thus, the correct response that demonstrates the Associative Property of Addition is:

\[ \boxed{(x + y) + z + r = x + (y + z) + r} \]

Answer 8

Which of the following demonstrates the Associative Property of Multiplication?
Responses:
(3⋅5)⋅2=3⋅(5⋅2)
(3⋅5)+2=(3+5)⋅2
(3+5)⋅2=3+(5⋅2)
(3⋅5)⋅2=(3⋅5)⋅2

Answer 9

The Associative Property of Multiplication states that the way in which factors are grouped in multiplication does not change the product. In mathematical terms, for any numbers \(a\), \(b\), and \(c\):

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

Let’s examine each of the given options to see which one correctly represents the Associative Property of Multiplication:

1. \((3 \cdot 5) \cdot 2 = 3 \cdot (5 \cdot 2)\): This correctly demonstrates the Associative Property of Multiplication because it shows the regrouping of the multiplication terms.

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

3. \((3 + 5) \cdot 2 = 3 + (5 \cdot 2)\): This expression also mixes addition and multiplication in a way that does not represent the Associative Property of Multiplication.

4. \((3 \cdot 5) \cdot 2 = (3 \cdot 5) \cdot 2\): This is simply a restatement of the same expression and does not indicate any grouping change.

Given this analysis, the correct response that demonstrates the Associative Property of Multiplication is:

\[ \boxed{(3 \cdot 5) \cdot 2 = 3 \cdot (5 \cdot 2)} \]

Answer 10

Use the Associative Property to determine which expressions are equivalent.Responses:
−3⋅(4x⋅−2)⋅−6y=(−3⋅4x)(−2⋅−6y)
−3⋅(4x⋅−2)⋅−6y=18−8x
−3⋅(4x⋅−2)⋅−6y=(−3⋅4x)⋅−2−6y
−3⋅(4x⋅−2)⋅−6y=−7xy

Answer 11

To evaluate which expressions are equivalent using the Associative Property, let's analyze each of the given expressions step by step.

We start with the expression: \[ -3 \cdot (4x \cdot -2) \cdot -6y \]

Evaluating the Expressions:

1. \(-3 \cdot (4x \cdot -2) \cdot -6y = (-3 \cdot 4x)(-2 \cdot -6y)\)

   Using the Associative Property:

   • By the Associative Property of Multiplication, we can regroup the factors.
   • This expression is valid; it shows how we can group the multiplication differently.

   Conclusion: This expression is equivalent.

2. \(-3 \cdot (4x \cdot -2) \cdot -6y = 18 - 8x\)

   We should simplify the left side:

   • Calculate \(4x \cdot -2 = -8x\).
   • Then, substituting that back in: \[ -3 \cdot (-8x) \cdot -6y = (-3 \cdot -8x) \cdot -6y = 24xy \cdot -6y = -144xy \]
   • The left side simplifies to \(-144xy\), not \(18 - 8x\).

   Conclusion: This expression is not equivalent.

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

   Simplifying the left side again:

   • As calculated earlier, this becomes \(-144xy\).
   • The right side is not a correct re-grouping. It seems to mix addition and multiplication incorrectly and does not apply the Associative Property.

   Conclusion: This expression is not equivalent.

4. \(-3 \cdot (4x \cdot -2) \cdot -6y = -7xy\)

   As established, the left side simplifies to \(-144xy\).

   • Therefore, this expression is not equivalent to \(-7xy\).

   Conclusion: This expression is not equivalent.

Summary of Conclusions:

• The first expression: \[ -3 \cdot (4x \cdot -2) \cdot -6y = (-3 \cdot 4x)(-2 \cdot -6y) \] is equivalent.
• The other expressions do not hold true upon simplification.

Thus, the only correct response is: \[ \boxed{-3 \cdot (4x \cdot -2) \cdot -6y = (-3 \cdot 4x)(-2 \cdot -6y)} \]