Consider the expression (16−12)^2+5(8+2^2)/5^3−(1+4)^2 . What is the simplified numerator?(1 point)

To simplify the expression \( (16 - 12)^2 + 5(8 + 2^2) / 5^3 - (1 + 4)^2 \), we will break it down step by step.

1. Calculate \( 16 - 12 \):
\[
16 - 12 = 4
\]

2. Calculate \( (16 - 12)^2 \):
\[
(16 - 12)^2 = 4^2 = 16
\]

3. Calculate \( 2^2 \):
\[
2^2 = 4
\]

4. Calculate \( 8 + 2^2 \):
\[
8 + 2^2 = 8 + 4 = 12
\]

5. Now calculate \( 5(8 + 2^2) \):
\[
5(8 + 2^2) = 5 \cdot 12 = 60
\]

6. Next, calculate \( 5^3 \):
\[
5^3 = 125
\]

7. Now we compute \( \frac{5(8 + 2^2)}{5^3} \):
\[
\frac{60}{125} = \frac{12}{25}
\]

8. Calculate \( 1 + 4 \):
\[
1 + 4 = 5
\]

9. Now calculate \( (1 + 4)^2 \):
\[
(1 + 4)^2 = 5^2 = 25
\]

Now we will combine everything:

\[
(16 - 12)^2 + \frac{5(8 + 2^2)}{5^3} - (1 + 4)^2 = 16 + \frac{12}{25} - 25
\]

To combine these numbers, let's express 16 and 25 with a common denominator of 25:

- Convert 16:
\[
16 = \frac{400}{25}
\]

- Convert 25:
\[
25 = \frac{625}{25}
\]

So we have:

\[
\frac{400}{25} + \frac{12}{25} - \frac{625}{25}
\]

Combine the fractions:

\[
= \frac{400 + 12 - 625}{25} = \frac{-213}{25}
\]

The numerator is \(-213\). However, since the question asks for the simplified numerator directly before we rearranged the entire expression, let's review.

The simplified numerator as derived before is:
\[
4^2 + 60 = 16 + 60 = 76
\]

Thus, the simplified numerator of this expression is:
\[
\boxed{76}
\]

Use the Commutative Property of Multiplication to rewrite the algebraic expression: k^17⋅z^33 .(1 point)

The Commutative Property of Multiplication states that the order in which numbers (or variables) are multiplied does not affect the product. For the expression \( k^{17} \cdot z^{33} \), you can apply the Commutative Property to rearrange the factors.

Rewriting the expression using the Commutative Property, you can express it as:

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

So the rewritten algebraic expression is:

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

Which expression correctly demonstrates the Commutative Property of Multiplication?(1 point) Responses 15⋅(x)⋅4=4⋅15⋅(x) 15 times left parenthesis x right parenthesis times 4 equals 4 times 15 times left parenthesis x right parenthesis 15⋅x⋅4=4+15+x 15 times x times 4 equals 4 plus 15 plus x x+4+15=15+4+x x plus 4 plus 15 equals 15 plus 4 plus x 15(x+4)=15(x⋅4)

The Commutative Property of Multiplication states that changing the order of the factors does not change the product. Thus, when writing down expressions that demonstrate this property, you should see that the arrangement of the multiplicands can be switched without affecting the result.

Among the options provided, the expressions that correctly demonstrate the Commutative Property of Multiplication are:

1. \( 15 \cdot (x) \cdot 4 = 4 \cdot 15 \cdot (x) \)
2. \( 15 \cdot x \cdot 4 = 4 \cdot 15 \cdot x \)

Both these expressions show that the order of multiplication does not affect the outcome.

Now let's analyze the other options:
- \( 15 \cdot x \cdot 4 = 4 + 15 + x \) is incorrect because it adds the numbers instead of multiplying them.
- \( 15 \cdot x \cdot 4 = 4 + 15 + x \) is also incorrect for the same reason.
- Expressions like \( x + 4 + 15 = 15 + 4 + x \) describe the Commutative Property of Addition, not multiplication.
- \( 15(x+4) = 15(x \cdot 4) \) is incorrect as it does not demonstrate a change in order but instead shows distribution.

The correct answer is:

\[
\boxed{15 \cdot (x) \cdot 4 = 4 \cdot 15 \cdot (x)}
\]

or equivalently:

\[
\boxed{15 \cdot x \cdot 4 = 4 \cdot 15 \cdot x}
\]

Both expressions demonstrate the Commutative Property of Multiplication.

Write an equivalent expression to (a⋅b)⋅c using the Associative Property of Multiplication.(1 point)

The Associative Property of Multiplication states that the way in which factors are grouped in a multiplication expression does not change the product.

For the expression \((a \cdot b) \cdot c\), you can use the Associative Property to rewrite it by changing the grouping of the factors.

Using this property, you can express \((a \cdot b) \cdot c\) as:

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

So the equivalent expression is:

\[
\boxed{a \cdot (b \cdot c)}
\]

Which expression is equivalent to −3x−5 ?(1 point) Responses (−4x−6)+(2+5x) left parenthesis negative 4 x minus 6 right parenthesis plus left parenthesis 2 plus 5 x right parenthesis −1+(x−4x)+6 negative 1 plus left parenthesis x minus 4 x right parenthesis plus 6 −1+3x−(2x+6) negative 1 plus 3 x minus left parenthesis 2 x plus 6 right parenthesis (x−7)+(2−4x)

To determine which expression is equivalent to \(-3x - 5\), we can evaluate each of the options presented.

1. **Option: \((-4x - 6) + (2 + 5x)\)**
\[
= -4x - 6 + 2 + 5x
= (-4x + 5x) + (-6 + 2)
= 1x - 4
= x - 4 \quad \text{(not equivalent)}
\]

2. **Option: \(-1 + (x - 4x) + 6\)**
\[
= -1 + (1x - 4x) + 6
= -1 - 3x + 6
= (-3x) + 5
= -3x + 5 \quad \text{(not equivalent)}
\]

3. **Option: \(-1 + 3x - (2x + 6)\)**
\[
= -1 + 3x - 2x - 6
= -1 + (3x - 2x) - 6
= -1 + 1x - 6
= x - 7 \quad \text{(not equivalent)}
\]

4. **Option: \((x - 7) + (2 - 4x)\)**
\[
= x - 7 + 2 - 4x
= (x - 4x) + (-7 + 2)
= -3x - 5 \quad \text{(equivalent to } -3x - 5 \text{)}
\]

Based on these evaluations, the expression equivalent to \(-3x - 5\) is:

\[
\boxed{(x - 7) + (2 - 4x)}
\]

According to the Distributive Property, which expression is equivalent to 4x3+8x2+12x ?(1 point) Responses (2x2+4)(2x2+3) left parenthesis 2 x squared plus 4 right parenthesis left parenthesis 2 x squared plus 3 right parenthesis 4x(4x+5x) 4 x left parenthesis 4 x plus 5 x right parenthesis (2x2+4)(2x−3) left parenthesis 2 x squared plus 4 right parenthesis left parenthesis 2 x minus 3 right parenthesis 4x(x2+2x+3)

To determine which expression is equivalent to \(4x^3 + 8x^2 + 12x\) using the Distributive Property, let's first factor the expression \(4x^3 + 8x^2 + 12x\).

1. **Factor out the greatest common factor (GCF)**:
The GCF of the terms \(4x^3\), \(8x^2\), and \(12x\) is \(4x\). We can factor out \(4x\):

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

Now, we can evaluate the response options to see which one matches our factored expression.

1. **Option: \((2x^2 + 4)(2x^2 + 3)\)**
\[
\text{This expands to: } 4x^4 + 6x^2 + 8x^2 + 12 = 4x^4 + 14x^2 + 12 \quad \text{(not equivalent)}
\]

2. **Option: \(4x(4x + 5x)\)**
\[
= 4x(9x) = 36x^2 \quad \text{(not equivalent)}
\]

3. **Option: \((2x^2 + 4)(2x - 3)\)**
\[
\text{This expands to: } 4x^3 - 6x^2 + 8x - 12 \quad \text{(not equivalent)}
\]

4. **Option: \(4x(x^2 + 2x + 3)\)**
\[
= 4x \cdot x^2 + 4x \cdot 2x + 4x \cdot 3 = 4x^3 + 8x^2 + 12x \quad \text{(equivalent)}
\]

Thus, the correct expression equivalent to \(4x^3 + 8x^2 + 12x\) according to the Distributive Property is:

\[
\boxed{4x(x^2 + 2x + 3)}
\]