Asked by Yes.

Use the Commutative Property of Multiplication to rewrite the algebraic expression: k^17⋅z^33
Answered by Yes.
Which expression correctly demonstrates the Commutative Property of Multiplication?(1 point)
Responses:
15⋅(x)⋅4=4⋅15⋅(x)
15⋅x⋅4=4+15+x
15(x+4)=15(x⋅4)
x+4+15=15+4+x
Answered by Yes.
Write an equivalent expression to (a⋅b)⋅c
using the Associative Property of Multiplication.
Answered by Yes.
Which expression is equivalent to −3x−5
Responses:
(−4x−6)+(2+5x)
−1+3x−(2x+6)
−1+(x−4x)+6
(x−7)+(2−4x)
Answered by Yes.
According to the Distributive Property, which expression is equivalent to 4x^3+8x^2+12x
Responses:
4x(x2+2x+3)
(2x^2+4)(2x−3)
(2x^2+4)(2x&2+3)
4x(4x+5x)
Answered by Yes.
Which of the following simplifies to the same expression as 4x^2−3x^2+5x−24
Responses:
2(x−12)
(x+8)(x−3)
4x(x−6)
(x+6)(x−4)
Answered by GPT-4o mini
According to the Commutative Property of Multiplication, the order of factors can be changed without affecting the product. Therefore, you can rewrite the expression \( k^{17} \cdot z^{33} \) as:

\[
z^{33} \cdot k^{17}
\]

Both expressions represent the same product.
Answered by GPT-4o mini
The expression that correctly demonstrates the Commutative Property of Multiplication is:

**15⋅(x)⋅4 = 4⋅15⋅(x)**

This shows that the order of multiplication can be changed while the product remains the same. The other options do not represent the commutative property correctly.
Answered by GPT-4o mini
Using the Associative Property of Multiplication, you can regroup the factors in a multiplication expression without changing the product.

For the expression \((a \cdot b) \cdot c\), you can rewrite it by changing the grouping:

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

Both expressions, \((a \cdot b) \cdot c\) and \(a \cdot (b \cdot c)\), are equivalent.
Answered by GPT-4o mini
To determine which expression is equivalent to \(-3x - 5\), we can simplify each option.

1. **Option 1:** \((-4x - 6) + (2 + 5x)\)
\[
= -4x - 6 + 2 + 5x = (5x - 4x) + (-6 + 2) = x - 4
\]

2. **Option 2:** \(-1 + 3x - (2x + 6)\)
\[
= -1 + 3x - 2x - 6 = 3x - 2x - 1 - 6 = x - 7
\]

3. **Option 3:** \(-1 + (x - 4x) + 6\)
\[
= -1 + (1x - 4x) + 6 = -1 - 3x + 6 = -3x + 5
\]

4. **Option 4:** \((x - 7) + (2 - 4x)\)
\[
= x - 7 + 2 - 4x = (x - 4x) + (-7 + 2) = -3x - 5
\]

The only expression that simplifies to \(-3x - 5\) is **Option 4**: \((x - 7) + (2 - 4x)\).
Answered by GPT-4o mini
To determine which expression is equivalent to \(4x^3 + 8x^2 + 12x\) using the Distributive Property, we can factor the expression.

First, we can factor out the common term \(4x\):

\[
4x^3 + 8x^2 + 12x = 4x(x^2 + 2x + 3)
\]

Now, let's analyze the given response options:

1. **Option 1:** \(4x(x^2 + 2x + 3)\)
- This is equivalent to \(4x^3 + 8x^2 + 12x\).

2. **Option 2:** \((2x^2 + 4)(2x - 3)\)
- Let's expand it:
\[
= 2x^2 \cdot 2x + 2x^2 \cdot (-3) + 4 \cdot 2x + 4 \cdot (-3) = 4x^3 - 6x^2 + 8x - 12
\]
- This is **not** equivalent.

3. **Option 3:** \((2x^2 + 4)(2x^2 + 3)\)
- Let's expand it:
\[
= 2x^2 \cdot 2x^2 + 2x^2 \cdot 3 + 4 \cdot 2x^2 + 4 \cdot 3 = 4x^4 + 6x^2 + 8x + 12
\]
- This is **not** equivalent.

4. **Option 4:** \(4x(4x + 5x)\)
- This simplifies to:
\[
= 4x(9x) = 36x^2
\]
- This is **not** equivalent.

The correct choice is **Option 1:** \(4x(x^2 + 2x + 3)\), as it is the only expression that is equivalent to \(4x^3 + 8x^2 + 12x\) according to the Distributive Property.
Answered by GPT-4o mini
To determine which expression simplifies to the same expression as \(4x^2 - 3x^2 + 5x - 24\), let's first simplify \(4x^2 - 3x^2 + 5x - 24\):

1. Combine like terms:
\[
4x^2 - 3x^2 = 1x^2 \quad \text{(or simply } x^2\text{)}
\]
Resulting in:
\[
x^2 + 5x - 24
\]

Next, we can factor \(x^2 + 5x - 24\) to identify an equivalent expression:

To factor \(x^2 + 5x - 24\), we look for two numbers that multiply to \(-24\) and add up to \(5\). The numbers \(8\) and \(-3\) work:

\[
x^2 + 5x - 24 = (x + 8)(x - 3)
\]

Now, let’s check each response option:

1. **Option 1:** \(2(x - 12)\)
\[
= 2x - 24
\]
- This does not simplify to \(x^2 + 5x - 24\).

2. **Option 2:** \((x + 8)(x - 3)\)
- We just found that this simplifies to \(x^2 + 5x - 24\).

3. **Option 3:** \(4x(x - 6)\)
\[
= 4x^2 - 24x
\]
- This does not simplify to \(x^2 + 5x - 24\).

4. **Option 4:** \((x + 6)(x - 4)\)
\[
= x^2 - 4x + 6x - 24 = x^2 + 2x - 24
\]
- This also does not simplify to \(x^2 + 5x - 24\).

The only option that correctly simplifies to the same expression as \(4x^2 - 3x^2 + 5x - 24\) is **Option 2:** \((x + 8)(x - 3)\).