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
Answers
GPT-4o mini
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**
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**
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
GPT-4o mini
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**
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**
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
GPT-4o mini
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**
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**