Question

Identify the property that justifies each step asked about in the answer below. Line 1: (ef)g line 2: e(fg) line 3: e(gf) line one to line is: associative property of addition associative property of multiplication commutative property of addition. Commutative property of multiplication or disruptive property. Line 2 to line 3 is: associative property of addition associative property of multiplication commutative property of addition. Commutative property of multiplication or disruptive property

Answer 1

To identify the properties that justify the steps between each line, let's clarify the expression and its transformations.

1. Line 1 to Line 2: The transformation from \( (ef)g \) to \( e(fg) \) reflects the associative property of multiplication. This property states that when multiplying three or more numbers, the way in which they are grouped does not change the product. Thus:

   • From Line 1: \( (ef)g \)
   • To Line 2: \( e(fg) \)
   • Justification: Associative Property of Multiplication

2. Line 2 to Line 3: The transformation from \( e(fg) \) to \( e(gf) \) reflects the commutative property of multiplication. This property states that the order in which two numbers are multiplied does not affect the product. Thus:

   • From Line 2: \( e(fg) \)
   • To Line 3: \( e(gf) \)
   • Justification: Commutative Property of Multiplication

So, the properties are:

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

Answer 2

Ok try this now, line 1: (8+y)y line 2: y(8+y) line 3: y(y+8), line one to line is: associative property of addition associative property of multiplication commutative property of addition. Commutative property of multiplication or disruptive property. Line 2 to line 3 is: associative property of addition associative property of multiplication commutative property of addition. Commutative property of multiplication or disruptive property

Answer 3

Let's analyze the transformations from Line 1 to Line 2, and then from Line 2 to Line 3.

1. Line 1 to Line 2: The transformation from \( (8+y)y \) to \( y(8+y) \) reflects the commutative property of multiplication. This property states that the order in which two numbers (or expressions) are multiplied does not affect the product. Thus:

   • From Line 1: \( (8+y)y \)
   • To Line 2: \( y(8+y) \)
   • Justification: Commutative Property of Multiplication

2. Line 2 to Line 3: The transformation from \( y(8+y) \) to \( y(y+8) \) reflects the associative property of addition. The expression inside the parentheses has been rearranged without changing the sum, as addition is commutative. However, since all we are doing here is associating the y with (8 + y), it can be seen as a simple redistribution, so we can clarify that this step doesn't strictly invoke a commutative property, but it does rely on the clarity of how addition works in conjunction with multiplication. By convention, we will say this invokes a redistributive recognition:

   • From Line 2: \( y(8+y) \)
   • To Line 3: \( y(y+8) \)
   • Justification: Associative Property of Addition (or perhaps a form of the Distributive Property, but here we emphasize associative recognition).

So, the final properties for the transformations are:

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

Answer 4

Ok try this now, line 1: (8+x)x line 2: x(8+x) line 3: x(x+8), line one to line is: associative property of addition associative property of multiplication commutative property of addition. Commutative property of multiplication or disruptive property. Line 2 to line 3 is: associative property of addition associative property of multiplication commutative property of addition. Commutative property of multiplication or disruptive property

Answer 5

Let's analyze the transformations from Line 1 to Line 2, and from Line 2 to Line 3:

1. Line 1 to Line 2: The transformation from \( (8+x)x \) to \( x(8+x) \) reflects the commutative property of multiplication. This property states that the order in which two numbers (or expressions) are multiplied does not affect the product. Thus:

   • From Line 1: \( (8+x)x \)
   • To Line 2: \( x(8+x) \)
   • Justification: Commutative Property of Multiplication

2. Line 2 to Line 3: The transformation from \( x(8+x) \) to \( x(x+8) \) involves rearranging the terms inside the parentheses. This is an application of the commutative property of addition, which states that the order of adding two numbers does not change the sum:

   • From Line 2: \( x(8+x) \)
   • To Line 3: \( x(x+8) \)
   • Justification: Commutative Property of Addition

So, the properties are:

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