n is an integer.
`{:("Quantity A","Quantity B"),((-3)^(2n),(-3)^(2n+1)):}`

Options

A. Quantity A is greater.
B. Quantity B is greater.
C. The two quantities are equal.
D. The relationship cannot be determined from the infromation given.

Answer

To solve the problem, we need to compare the two quantities given:

**Quantity A:** \((-3)^{2n}\)
**Quantity B:** \((-3)^{2n+1}\)

### Step 1: Analyze Quantity A
We start with Quantity A, which is \((-3)^{2n}\).

Since \(2n\) is an even number (as it is a multiple of 2), we know that raising a negative number to an even power results in a positive number. Therefore, we can conclude:

\[ (-3)^{2n} > 0 \]

### Step 2: Analyze Quantity B
Next, we look at Quantity B, which is \((-3)^{2n+1}\).

Here, \(2n+1\) is an odd number. When a negative number is raised to an odd power, the result is negative. Thus, we have:

\[ (-3)^{2n+1} < 0 \]

### Step 3: Compare the Two Quantities
Now we can compare the two quantities we analyzed:

- Quantity A: \((-3)^{2n} > 0\)
- Quantity B: \((-3)^{2n+1} < 0\)

Since Quantity A is positive and Quantity B is negative, we conclude that:

\[ \text{Quantity A} > \text{Quantity B} \]

### Final Conclusion
Thus, the relationship between Quantity A and Quantity B is:

\[ \text{Quantity A} > \text{Quantity B} \]

The correct option is that Quantity A is greater than Quantity B.

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