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Which of the following correctly demonstrates the use of the Commutative Property of Multiplication?(1 point) Responses 2(b10+z11)=(b10+z11)⋅2 2 left parenthesis b superscript 10 baseline plus z superscript 11 baseline right parenthesis equals left parenthesis b superscript 10 baseline plus z superscript 11 baseline right parenthesis times 2 11(b+z)=11b+11z 11 left parenthesis b plus z right parenthesis equals 11 b plus 11 z 3(b10+4)=3(4+b10)3 3 left parenthesis b superscript 10 baseline plus 4 right parenthesis equals 3 left parenthesis 4 plus b superscript 10 baseline right parenthesis 3 z11⋅2=z11+2

11 answers

The Commutative Property of Multiplication states that changing the order of the factors does not change the product. In other words, \( a \times b = b \times a \) for any numbers \( a \) and \( b \).

Let's analyze the options provided:

  1. \( 2(b^{10} + z^{11}) = (b^{10} + z^{11}) \cdot 2 \)

    • This illustrates the Associative Property of Multiplication (because it shows grouping of factors) and the Distributive Property (distributing the multiplication over addition), but not the Commutative Property.

  2. \( 11(b + z) = 11b + 11z \)

    • This shows the Distributive Property, as it distributes the multiplication over addition. This is not the Commutative Property.

  3. \( 3(b^{10} + 4) = 3(4 + b^{10}) \)

    • This correctly demonstrates the Commutative Property because it shows that \( b^{10} + 4 \) is equal to \( 4 + b^{10} \) due to the property of addition, and the multiplication remains unaffected.

  4. \( 3z^{11} \cdot 2 = z^{11} + 2 \)

    • This expression is incorrect in terms of multiplication.

Therefore, the correct answer demonstrating the Commutative Property of Multiplication is option 3:

\( 3(b^{10} + 4) = 3(4 + b^{10}) \).

Which of the following correctly demonstrates the Commutative Property of Addition?(1 point) Responses abc+xyz=xyz+abc a b c plus x y z equals x y z plus a b c 123+456=579 123 plus 456 equals 579 abcxyz=zyxcba a b c x y z equals z y x c b a 0+579=579 0 plus 579 equals 579

The Commutative Property of Addition states that changing the order of the addends does not change the sum. In other words, \( a + b = b + a \) for any numbers \( a \) and \( b \).

Now, let's analyze the provided options:

  1. \( abc + xyz = xyz + abc \)

    • This correctly demonstrates the Commutative Property of Addition, as it shows that the order of the terms can be switched without changing the sum.

  2. \( 123 + 456 = 579 \)

    • This is a true statement about the sum of 123 and 456, but it does not demonstrate the Commutative Property of Addition explicitly.

  3. \( abcxyz = zyxabc \)

    • This expression does not relate to addition; it seems to show multiplication or juxtaposition of terms, and thus does not relate to the Commutative Property of Addition.

  4. \( 0 + 579 = 579 \)

    • This is a true statement about adding zero to 579, but it does not demonstrate the Commutative Property of Addition.

The correct answer demonstrating the Commutative Property of Addition is:

\( abc + xyz = xyz + abc \).

Use the Commutative Property to determine the missing step in proving the equivalence of 12a+10+a=10+a+12a. Step 1: [missing] Step 2: 10+13a=10+13a Step 3: The expressions are equivalent because they both equal the same value. (1 point) Responses 12a+10+a−a=10+a+12a−a d plus 10 plus a equals 10 plus a plus d 12a+10=10+12a d plus 10 plus a equals 10 plus a plus d 10+12a+a=10+13a d plus 10 plus a equals 10 plus a plus d 12a+10+a−10=10+a+12a−10

To fill in the missing step, we need to use the Commutative Property of Addition, which states that the order of adding numbers can be changed without affecting the sum.

Here's the current progress in the proof:

  • Step 1: [missing]
  • Step 2: \( 10 + 13a = 10 + 13a \)
  • Step 3: The expressions are equivalent because they both equal the same value.

Let's analyze the choices based on what we need:

  1. \( 12a + 10 + a - a = 10 + a + 12a - a \)

    • This does not demonstrate the Commutative Property clearly; it looks like it's reducing terms unnecessarily.

  2. \( d + 10 + a = 10 + a + d \)

    • This statement is true due to the Commutative Property but uses a generic variable \(d\), so it's not relevant to \(12a\) and \(10\).

  3. \( 12a + 10 = 10 + 12a \)

    • This correctly demonstrates the Commutative Property of Addition and directly shows that \(12a + 10\) can be reordered.

  4. \( d + 10 + a = 10 + a + d \)

    • This is another instance of the Commutative Property but uses a generic variable, so it's not relevant to this context.

  5. \( 12a + 10 + a - 10 = 10 + a + 12a - 10 \)

    • This does not clearly utilize the Commutative Property, as it is reducing terms incorrectly.

Based on the need to use the Commutative Property, the appropriate missing step should be:

\( 12a + 10 = 10 + 12a \)

So, the complete sequence would look like this:

  • Step 1: \( 12a + 10 = 10 + 12a \) (using the Commutative Property)
  • Step 2: \( 10 + 13a = 10 + 13a \)
  • Step 3: The expressions are equivalent because they both equal the same value.
Substituting 1 for x in the equation 5x + 3=x⋅5 + 3 is a test case for which property?(1 point) Responses The Commutative Property of Multiplication The Commutative Property of Multiplication The Associative Property of Multiplication The Associative Property of Multiplication The Commutative Property of Addition The Commutative Property of Addition The Associative Property of Addition The Associative Property of Addition

To understand which property is being tested by substituting \(1\) for \(x\) in the equation \(5x + 3 = x \cdot 5 + 3\), let's examine the equation more closely.

When we substitute \(1\) for \(x\):

  1. The left side becomes: \[ 5(1) + 3 = 5 + 3 = 8 \]

  2. The right side becomes: \[ 1 \cdot 5 + 3 = 5 + 3 = 8 \]

Both sides equal \(8\), confirming that the equation holds true for \(x = 1\).

The property being tested by this substitution is related to the equality of the two expressions when \(x=1\), specifically showing that the multiplication of \(x\) by \(5\) can be rearranged, representing a test of equality under substitution.

However, the equation also emphasizes that \(5x\) and \(x \cdot 5\) are equivalent, which is a direct application of the Commutative Property of Multiplication, since it demonstrates that the order in which multiplication is performed does not affect the result.

Given the available responses, the answer to which property is being tested is:

The Commutative Property of Multiplication.

Which step contains an error using the Commutative Property of Addition in verifying 8+x2+17x−x2+4=10+4x2+10x−4x2+2+7x? Step 1: 8+4+x2−x2+17x=10+2+10x−4x2+7x +4x2 Step 2: 8+4+17x+x2−x2=10+2+4x2−4x2+10x+7x Step 3: 12+17x+x2=12−x2+17x (1 point) Responses Step 2 Step 2 No mistake was made. No mistake was made. Step 3 Step 3 Step 1 Step 1

To identify the step that contains an error using the Commutative Property of Addition in the provided equation verification, let's carefully analyze each step. The goal is to simplify both sides and check for correctness.

Steps Provided:

  • Step 1: \( 8 + 4 + x^2 - x^2 + 17x = 10 + 2 + 10x - 4x^2 + 7x + 4x^2 \)
  • Step 2: \( 8 + 4 + 17x + x^2 - x^2 = 10 + 2 + 4x^2 - 4x^2 + 10x + 7x \)
  • Step 3: \( 12 + 17x + x^2 = 12 - x^2 + 17x \)

Analysis of Each Step:

  1. Step 1:

    • Left: \( 8 + 4 + x^2 - x^2 + 17x \) simplifies correctly to \( 12 + 17x \).
    • Right: \( 10 + 2 + 10x - 4x^2 + 7x + 4x^2 \) simplifies to \( 12 + 17x \) as well.
    • No errors detected in this step.

  2. Step 2:

    • Left: \( 8 + 4 + 17x + x^2 - x^2 \) simplifies correctly to \( 12 + 17x \).
    • Right: Let's simplify \( 10 + 2 + 4x^2 - 4x^2 + 10x + 7x \):
      • \( 10 + 2 = 12 \)
      • \( 4x^2 - 4x^2 = 0 \)
      • \( 10x + 7x = 17x \)
    • The right side also simplifies to \( 12 + 17x \). No errors here as well.

  3. Step 3:

    • This states \( 12 + 17x + x^2 = 12 - x^2 + 17x \).
    • On the left, we have \( 12 + 17x + x^2 \).
    • On the right, we have \( 12 - x^2 + 17x \).
    • The two sides are not equivalent due to the presence of \( +x^2 \) on the left and \( -x^2 \) on the right.

Conclusion:

The error occurs in Step 3 where the equation is incorrectly stated as equal.

Thus, the correct answer is:

Step 3 contains the error.

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 plus 8 plus 3 (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⋅3)

The Associative Property states that the way in which numbers are grouped in addition or multiplication does not affect the result. Specifically, it applies as follows:

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

To determine which of the provided expressions could be rewritten using the Associative Property, we need to look for expressions that contain at least three terms or factors.

Let's analyze the choices:

  1. \((9 \cdot 5) + 8 + 3\)

    • This expression consists of three terms. We could group them in different ways, such as \( ((9 \cdot 5) + 8) + 3 \) or \( (9 \cdot 5) + (8 + 3) \). This can be rewritten using the Associative Property.

  2. \((5 + 8) \cdot 3\)

    • This expression consists of only two terms being multiplied. The Associative Property does not apply here because it requires grouping of three or more elements. This cannot be rewritten using the Associative Property.

  3. \((5 \cdot 8) + 3\)

    • This expression consists of two terms. The Associative Property does not apply here because it requires grouping of three or more elements. This cannot be rewritten using the Associative Property.

  4. \((9 \cdot 5) \cdot (8 \cdot 3)\)

    • This expression is a product of two terms. The Associative Property applies in multiplication, but here we can't regroup two entire products since they are not in the same operation (they can't be grouped while still maintaining their original meaning). This cannot be rewritten using the Associative Property.

Conclusion:

The expression that can be rewritten using the Associative Property is:

\((9 \cdot 5) + 8 + 3\).