Select the open phrase for each of the following:

Thrice the difference of the integer N and 4.*

4-3N

3N-4

3(N-4)

3(4-N)

Four  more than six times the integer N*

6(N+4)

6N+4

4N+6

4(N+6)

The cube of the difference of two numbers N and P*

N^3 - P^3

(N^3 x P^3)

(N x P)^3

(N - P)^3

Let's analyze each of the given statements and select the correct open phrase:

1. **Thrice the difference of the integer \( N \) and 4**
   - "Thrice" means three times, and "the difference of \( N \) and 4" means \( N - 4 \).
   - So, this translates to \( 3(N - 4) \).
   - **Correct Answer:** **3(N - 4)**

2. **Four more than six times the integer \( N \)**
   - "Six times the integer \( N \)" means \( 6N \), and "four more than" means we add 4 to this.
   - So, this translates to \( 6N + 4 \).
   - **Correct Answer:** **6N + 4**

3. **The cube of the difference of two numbers \( N \) and \( P \)**
   - "The difference of \( N \) and \( P \)" means \( N - P \), and "the cube of" means raising this difference to the third power.
   - So, this translates to \( (N - P)^3 \).
   - **Correct Answer:** **(N - P)^3**

### Summary:
1. **Thrice the difference of the integer \( N \) and 4:** 3(N - 4)
2. **Four more than six times the integer \( N \):** 6N + 4
3. **The cube of the difference of two numbers \( N \) and \( P \):** (N - P)^3

Would you like more details on any of these open phrases, or do you have any questions?

Here are 5 related questions:
1. How would you express "twice the sum of \( N \) and 7" in algebraic form?
2. What is the difference between a phrase like "thrice" and "three times" in mathematical expressions?
3. How do you represent "five less than four times \( N \)" algebraically?
4. Can you explain how to translate "the square of a sum" into an expression?
5. What are some common mistakes when translating verbal phrases into algebraic expressions?

**Tip:** When translating word phrases to algebraic expressions, carefully identify the operations (like addition, subtraction, multiplication) and their order to avoid errors.

Explore algebraic expressions with examples and detailed solutions. Learn how to translate verbal phrases into algebraic equations, covering concepts like cubes of differences and multiplication with integers. Suitable for high school students.

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