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Identify the property that justifies each step asked about in the answer area below.

Line 1: , left parenthesis, a, c, right parenthesis, b
Line 1: =
(ac)b
Line 2: , a, left parenthesis, c, b, right parenthesis
Line 2: =
a(cb)
Line 3: , a, left parenthesis, b, c, right parenthesis
Line 3: =
a(bc)
Answer
Attempt 1 out of 2

Line 1 to Line 2:
Line 2 to Line 3:

11 answers

Let's identify the properties that justify the steps from Line 1 to Line 2 and from Line 2 to Line 3.

Line 1 to Line 2: The transformation from \((ac)b\) to \(a(cb)\) uses the Associative Property of Multiplication. This property allows us to regroup the factors in a multiplication without changing the product.

Line 2 to Line 3: The transformation from \(a(cb)\) to \(a(bc)\) also uses the Associative Property of Multiplication. Just like the first transformation, it allows us to regroup the multiplication.

So, the complete answer is:

  • Line 1 to Line 2: Associative Property of Multiplication
  • Line 2 to Line 3: Associative Property of Multiplication
1 of those were rong

I apologize for the oversight. Let's clarify the steps again.

Line 1 to Line 2: From \((ac)b\) to \(a(cb)\), we apply the Commutative Property of Multiplication. The factors \(ac\) can be rearranged to \(a(c)\) because multiplication is commutative. (You could also say it depends on how you parenthesize the multiplication, but the primary property here is commutation.)

Line 2 to Line 3: From \(a(cb)\) to \(a(bc)\), we again apply the Commutative Property of Multiplication. The factors within the parentheses are being rearranged.

So the correct answer is:

  • Line 1 to Line 2: Commutative Property of Multiplication
  • Line 2 to Line 3: Associative Property of Multiplication

Thank you for your understanding.

wrong aagian you had them flipped

I apologize for the confusion earlier. Let’s clarify the justification for each step accurately:

  1. Line 1 to Line 2:

    • The step from \((ac)b\) to \(a(cb)\) involves rearranging the multiplication inside the parentheses. This uses the Associative Property of Multiplication. The parentheses change how we group the multiplication without changing the overall product.

  2. Line 2 to Line 3:

    • The transformation from \(a(cb)\) to \(a(bc)\) uses the Commutative Property of Multiplication. This property states that the order of multiplication does not affect the product, allowing us to rearrange the terms inside the parentheses.

So the correct designations should be:

  • Line 1 to Line 2: Associative Property of Multiplication
  • Line 2 to Line 3: Commutative Property of Multiplication

Thank you for your patience as we sorted this out!

Identify the property that justifies each step asked about in the answer area below.
Line 1: , left parenthesis, 4, x, right parenthesis, left parenthesis, 5, y, right parenthesis
Line 1: =
(4x)(5y)
Line 2: , 4, dot, left parenthesis, x, dot, 5, right parenthesis, dot, y
Line 2: =
4⋅(x⋅5)⋅y
Line 3: , 4, dot, left parenthesis, 5, dot, x, right parenthesis, dot, y
Line 3: =
4⋅(5⋅x)⋅y
Line 4: , left parenthesis, 4, dot, 5, right parenthesis, left parenthesis, x, dot, y, right parenthesis
Line 4: =
(4⋅5)(x⋅y)
Line 5: , 20, x, y
Line 5: =
20xy
Answer
Attempt 1 out of 2

Line 1 to Line 2:
Line 2 to Line 3:
Line 3 to Line 4:

Let’s identify the properties that justify each step for the transformation from each line:

  1. Line 1 to Line 2:

    • The step from \((4x)(5y)\) to \(4 \cdot (x \cdot 5) \cdot y\) uses the Associative Property of Multiplication. This property allows us to regroup the factors, although technically, this could also be viewed as separating the individual constants from the variables.

  2. Line 2 to Line 3:

    • The step from \(4 \cdot (x \cdot 5) \cdot y\) to \(4 \cdot (5 \cdot x) \cdot y\) uses the Commutative Property of Multiplication. This property allows us to change the order of multiplication inside the parentheses.

  3. Line 3 to Line 4:

    • The step from \(4 \cdot (5 \cdot x) \cdot y\) to \((4 \cdot 5)(x \cdot y)\) uses the Associative Property of Multiplication again. We are regrouping the terms to factor out the constants and the variables.

So, the complete response is:

  • Line 1 to Line 2: Associative Property of Multiplication
  • Line 2 to Line 3: Commutative Property of Multiplication
  • Line 3 to Line 4: Associative Property of Multiplication

Thank you for your patience!

Are you sure thats correct

Let's carefully revisit the transformations and identify the properties accurately step by step.

  1. Line 1 to Line 2:

    • The transition from \((4x)(5y)\) to \(4 \cdot (x \cdot 5) \cdot y\) can be justified by the Associative Property of Multiplication, as it involves regrouping the factors while separating the constants. Here, we regard the multiplication structure more explicitly.

  2. Line 2 to Line 3:

    • The transition from \(4 \cdot (x \cdot 5) \cdot y\) to \(4 \cdot (5 \cdot x) \cdot y\) uses the Commutative Property of Multiplication since we are rearranging the factors within the parentheses.

  3. Line 3 to Line 4:

    • The transition from \(4 \cdot (5 \cdot x) \cdot y\) to \((4 \cdot 5)(x \cdot y)\) also uses the Associative Property of Multiplication. Here, we group the constants together and the variables together.

So, the correct properties are:

  • Line 1 to Line 2: Associative Property of Multiplication
  • Line 2 to Line 3: Commutative Property of Multiplication
  • Line 3 to Line 4: Associative Property of Multiplication

If you believe there are mistakes, I appreciate your corrections as we want to ensure the understanding of the properties employed is accurate. Thank you for your diligence.

Identify the property that justifies each step asked about in the answer area below.
Line 1: , left parenthesis, minus, 8, plus, 4, x, right parenthesis, plus, 9
Line 1: =
(−8+4x)+9
Line 2: , left parenthesis, 4, x, plus, left parenthesis, minus, 8, right parenthesis, right parenthesis, plus, 9
Line 2: =
(4x+(−8))+9
Line 3: , 4, x, plus, left parenthesis, minus, 8, plus, 9, right parenthesis
Line 3: =
4x+(−8+9)
Line 4: , 4, x, plus, 1
Line 4: =
4x+1
Answer
Attempt 1 out of 2

Line 1 to Line 2:
Line 2 to Line 3:

Let’s analyze each step carefully and identify the properties involved.

  1. Line 1 to Line 2:

    • The transition from \((−8 + 4x) + 9\) to \((4x + (−8)) + 9\) uses the Commutative Property of Addition. This property allows us to change the order of the terms in the sum, rearranging \(-8\) and \(4x\) while maintaining the same expression.

  2. Line 2 to Line 3:

    • The transition from \((4x + (−8)) + 9\) to \(4x + (−8 + 9)\) uses the Associative Property of Addition. This property permits us to regroup the additive terms, moving the parentheses around to change the grouping without altering the sum.

So the correct identification of properties for these two steps is:

  • Line 1 to Line 2: Commutative Property of Addition
  • Line 2 to Line 3: Associative Property of Addition

If you have additional queries or need further clarification, feel free to ask!