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Let's solve the problems one by one from the image you uploaded:

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**1. What is \(3(4 - 5) - (9 + 12)\)?**

- First, simplify inside the parentheses:
  \[
  3(4 - 5) = 3(-1) = -3
  \]
  \[
  9 + 12 = 21
  \]
- Now, subtract:
  \[
  -3 - 21 = -24
  \]
  So, the answer is **(a) -24**.

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**2. Simplify \(\frac{n^x \cdot n^x}{(n + n)^x}\)**

- Simplify the denominator:
  \[
  n + n = 2n
  \]
  So, the expression becomes:
  \[
  \frac{n^{2x}}{(2n)^x}
  \]
  Simplify this:
  \[
  \frac{n^{2x}}{2^x \cdot n^x} = \frac{n^{2x}}{n^x \cdot 2^x} = \frac{n^{2x}}{n^x \cdot 2^x} = \frac{n^x}{2^x} = \left(\frac{n}{2}\right)^x
  \]
  Therefore, the simplified expression is **(d) \(\left(\frac{n}{2}\right)^x\)**.

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**3. Set \( N \) has 52 elements, set \( M \) has 21 elements, and set \( C \) is the combination of sets \( N \) and \( M \) with 65 elements. How many elements does set \( P \) have if it is the intersection of set \( N \) and set \( M \)?**

Using the formula for the union of two sets:
\[
|N \cup M| = |N| + |M| - |N \cap M|
\]
We know:
\[
|N \cup M| = 65, |N| = 52, |M| = 21
\]
Substitute:
\[
65 = 52 + 21 - |N \cap M|
\]
Solve for \( |N \cap M| \):
\[
65 = 73 - |N \cap M| \implies |N \cap M| = 73 - 65 = 8
\]
So, the answer is **(b) 8**.

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**4. What number is midway between \(5 - \frac{1}{4}\) and \(6 + \frac{7}{5}\)?**

- Convert the numbers to improper fractions:
  \[
  5 - \frac{1}{4} = \frac{20}{4} - \frac{1}{4} = \frac{19}{4}, \quad 6 + \frac{7}{5} = \frac{30}{5} + \frac{7}{5} = \frac{37}{5}
  \]
- Now, find the midpoint:
  \[
  \text{Midpoint} = \frac{\frac{19}{4} + \frac{37}{5}}{2}
  \]
  Find a common denominator:
  \[
  \frac{19}{4} = \frac{95}{20}, \quad \frac{37}{5} = \frac{148}{20}
  \]
  Add the fractions:
  \[
  \frac{95}{20} + \frac{148}{20} = \frac{243}{20}
  \]
  Divide by 2:
  \[
  \frac{243}{40}
  \]
  So, the answer is **(a) \( \frac{233}{40} \)**.

---

**5. Find the mean of the following data values: 16, 14, 23, 25, 29, 15, 14.**

- Add the values:
  \[
  16 + 14 + 23 + 25 + 29 + 15 + 14 = 136
  \]
- Divide by the number of values (7):
  \[
  \frac{136}{7} = 19.43
  \]
  So, the answer is **(c) 19.43**.

---

**6. Three exterior angles of a hexagon are 70°, 50°, and 80°. The remaining three exterior angles are in the ratio of 2:3:4. Find the largest of these three interior angles.**

- The sum of the exterior angles of any polygon is always 360°.
- Subtract the given angles from 360°:
  \[
  360 - (70 + 50 + 80) = 360 - 200 = 160
  \]
- Let the three unknown exterior angles be \(2x\), \(3x\), and \(4x\).
- Their sum is:
  \[
  2x + 3x + 4x = 9x
  \]
  So, \(9x = 160 \implies x = \frac{160}{9} \approx 17.78\).
  - The largest exterior angle is \(4x = 4 \times 17.78 \approx 71.11^\circ\).
  - The corresponding interior angle is:
  \[
  180^\circ - 71.11^\circ \approx 108.89^\circ
  \]
  So, the largest angle is **(d) 71.11**.

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Would you like further details on any of these? 

### Here are 5 related questions you might find interesting:
1. How do you find the midpoint between two numbers with fractions?
2. Can you explain the difference between the union and intersection of sets?
3. How can you calculate the volume of more complex prisms?
4. Why is the sum of exterior angles always 360 degrees for any polygon?
5. What is the significance of simplifying algebraic expressions?

**Tip**: When solving problems with multiple steps, always start by simplifying smaller components, such as parentheses or fractions, before tackling the full expression.

Learn how to simplify the expression 3(4 - 5) - (9 + 12) by using basic arithmetic and the order of operations. This problem is ideal for students in Grades 7-8.

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