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

Question

Line 1: , left parenthesis, 8, x, right parenthesis, left parenthesis, 10, y, right parenthesis
Line 1: =
(8x)(10y)
Line 2: , 8, dot, left parenthesis, x, dot, 10, right parenthesis, dot, y
Line 2: =
8⋅(x⋅10)⋅y
Line 3: , 8, dot, left parenthesis, 10, dot, x, right parenthesis, dot, y
Line 3: =
8⋅(10⋅x)⋅y
Line 4: , left parenthesis, 8, dot, 10, right parenthesis, left parenthesis, x, dot, y, right parenthesis
Line 4: =
(8⋅10)(x⋅y)
Line 5: , 80, x, y
Line 5: =
80xy

Answer 1

Here is the identified property that justifies each step:

1. Step 1 to Step 2:

   • Justification: Associative Property of Multiplication
   • Explanation: This property allows us to regroup the factors (in this case, separating \(8\) and \(10\) from \(x\) and \(y\)) without changing the overall product.

2. Step 2 to Step 3:

   • Justification: Commutative Property of Multiplication
   • Explanation: This property allows us to change the order of the factors. Here, \(x \cdot 10\) is rearranged to \(10 \cdot x\).

3. Step 3 to Step 4:

   • Justification: Associative Property of Multiplication
   • Explanation: This allows us to regroup the multiplying factors, moving \(10\) to be grouped with \(8\) and leaving \(x\) separated.

4. Step 4 to Step 5:

   • Justification: Multiplicative Identity and Simplification
   • Explanation: By multiplying \(8\) and \(10\), we obtain \(80\), and since multiplication of \(x\) and \(y\) is maintained, it results in \(80xy\).

Thus, each step is justified by these properties of multiplication.