y is an integer.
`{:("Quantity A","Quantity B"),((1)/(2^(y)),(1)/(3^(y))):}`

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:** \(\frac{1}{2^y}\)
**Quantity B:** \(\frac{1}{3^y}\)

where \(y\) is an integer.

### Step 1: Analyze the expressions for different values of \(y\)

1. **If \(y = 1\):**
- Quantity A: \(\frac{1}{2^1} = \frac{1}{2}\)
- Quantity B: \(\frac{1}{3^1} = \frac{1}{3}\)
- Comparison: \(\frac{1}{2} > \frac{1}{3}\)
- Conclusion: Quantity A is greater than Quantity B.

2. **If \(y = -1\):**
- Quantity A: \(\frac{1}{2^{-1}} = 2\)
- Quantity B: \(\frac{1}{3^{-1}} = 3\)
- Comparison: \(2 < 3\)
- Conclusion: Quantity B is greater than Quantity A.

3. **If \(y = 0\):**
- Quantity A: \(\frac{1}{2^0} = 1\)
- Quantity B: \(\frac{1}{3^0} = 1\)
- Comparison: \(1 = 1\)
- Conclusion: Quantity A is equal to Quantity B.

### Step 2: Generalize the comparison

- For positive values of \(y\):
- As \(y\) increases, \(2^y\) grows faster than \(3^y\), hence \(\frac{1}{2^y}\) will be greater than \(\frac{1}{3^y}\).

- For negative values of \(y\):
- As \(y\) decreases (becomes more negative), \(2^y\) approaches zero slower than \(3^y\), hence \(\frac{1}{3^y}\) will be greater than \(\frac{1}{2^y}\).

### Conclusion

Since the relationship between Quantity A and Quantity B changes depending on the value of \(y\) (whether it is positive, negative, or zero), we cannot determine a consistent relationship between the two quantities.

Thus, the answer is that the relationship cannot be determined from the information given.

### Final Answer:
**Option 4: The relationship cannot be determined from the information given.**

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