Question

For the following problem, match the name of the number property that was used to get to each step from the previous step.

37 · 2 · (-5 + 1/2 + 5)

1. 37 · 2 · (-5 + 5 + 1/2)
identity property of multiplication
2. 37 · 2 · (0 + 1/2)
inverse property of multiplication
3. 37 · 2 · 1/2
identity property of addition
4. 37 · 1
inverse property of addition
5. 37
commutative property

Answers

GPT-4o mini
Let’s match each step with the corresponding number property used in the given expression:

1. **37 · 2 · (-5 + 5 + 1/2)**
This step rearranges the terms in the parentheses. Since \(-5 + 5 = 0\), this demonstrates the **identity property of addition**.
- **Property Used:** identity property of addition

2. **37 · 2 · (0 + 1/2)**
Here, we are showing that adding 0 (the result of \(-5 + 5\)) to \(1/2\) does not change the value. This step demonstrates the **identity property of addition** again.
- **Property Used:** identity property of addition

3. **37 · 2 · 1/2**
This step follows from the fact that \(0 + 1/2\) is simply \(1/2\). Here, we are applying the **identity property of multiplication** because multiplying by \(1/2\) does not change the overall multiplication by \(2\).
- **Property Used:** identity property of multiplication

4. **37 · 1**
This step simplifies \(2 \cdot 1/2\) to \(1\), essentially canceling \(2\) out. Thus, it reflects the **inverse property of multiplication**, which states that a number multiplied by its reciprocal gives 1 (in this case, \(2 \cdot 1/2 = 1\)).
- **Property Used:** inverse property of multiplication

5. **37**
Here, the step simply removes the \(1\) because multiplying by \(1\) does not change the value. This is again demonstrating the **identity property of multiplication** (multiplying by 1 has no effect).
- **Property Used:** identity property of multiplication

Now summarizing the steps and matching them with the appropriate properties used:

1. **identity property of addition**
2. **identity property of addition**
3. **identity property of multiplication**
4. **inverse property of multiplication**
5. **identity property of multiplication**

These will carry you through all the simplifications performed stepwise on the original expression.

#### Final Matches:
1. identity property of addition
2. identity property of addition
3. identity property of multiplication
4. inverse property of multiplication
5. identity property of multiplication

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