Responses
11(b+z)=11b+11z
11 left parenthesis b plus z right parenthesis equals 11 b plus 11 z
z11⋅2=z11+2
z superscript 11 baseline times 2 equals z superscript 11 baseline plus 2
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
2(b10+z11)=(b10+z11)⋅2
9 answers
The Commutative Property of Multiplication states that changing the order of the factors does not change the product. The correct demonstration of the Commutative Property of Multiplication from your options is:
2(b^10 + z^11) = (b^10 + z^11)·2
This shows that multiplying the expression (b^10 + z^11) by 2 is the same as multiplying 2 by the expression (b^10 + z^11), which illustrates the commutativity of multiplication.
The Commutative Property of Addition states that changing the order of the addends does not change the sum. The correct demonstration of the Commutative Property of Addition from your options is:
abc + xyz = xyz + abc
This shows that the sum of the two expressions is the same regardless of the order in which they are added, illustrating the commutativity of addition.
To correctly demonstrate the use of the Commutative Property in proving the equivalence of \( 12a + 10 + a = 10 + a + 12a \), the missing step should show the rearrangement of the terms in the expression.
The correct missing step is:
Step 1: \( 10 + 12a + a = 10 + a + 12a \)
This uses the Commutative Property to reorder the terms. Now the expression can be simplified further in Step 2.
So, the response corresponding to the missing step is:
10 + 12a + a = 10 + 13a.
Substituting 1 for \( x \) in the equation \( 5x + 3 = x \cdot 5 + 3 \) serves as a test case to demonstrate the Commutative Property of Multiplication.
This property indicates that the order in which two numbers are multiplied does not affect the product, as shown in this equation where \( 5x \) and \( x \cdot 5 \) are equal due to this property. Therefore, the correct response is:
The Commutative Property of Multiplication.
To identify the error in the steps, let's analyze them one by one:
Step 1: \[ 8 + 4 + x^2 - x^2 + 17x = 10 + 2 + 10x - 4x^2 + 7x + 4x^2 \] This step appears to correctly rearrange and combine like terms on the left side and states the right side accurately.
Step 2: \[ 8 + 4 + 17x + x^2 - x^2 = 10 + 2 + 4x^2 - 4x^2 + 10x + 7x \] Here, \( x^2 - x^2 \) cancels out correctly and the left side becomes \( 12 + 17x \). However, on the right side, when combining \( 10x \) and \( 7x \), it should not involve cancellation of \( 4x^2 \) as it should be simplified to just a single term. Thus, the right side should contain \( 10 + 2 + 10x + 7x \) giving \( 12 + 17x \), instead of combining them incorrectly.
Step 3: \[ 12 + 17x + x^2 = 12 - x^2 + 17x \] In this final step, there seems to be no further logical flow since \( x^2 \) was not handled correctly before.
Therefore, the error occurs in Step 2, as it mismanages the simplification and combination of terms on the right side of the equation.
So, the correct response is:
Step 2.
