Which of the following relation (s) is/are false ?
I. `(27)^((1)/(3)) gt (13)^((1)/(2)) gt (47)^((1)/(6))`
II. `(23)^((1)/(3)) lt (49)^((1)/(2)) lt (52)^((1)/(6))`
III. `(52)^((1)/(6)) lt (41)^((1)/(3)) lt (37)^((1)/(2))`

Options

A. Only III
B. Both I and II
C. II and III only
D. All relations are false

Answer

To determine which of the given relations is false, we will evaluate each statement one by one by comparing the values of the expressions involved. We will use the method of finding the least common multiple (LCM) of the denominators of the fractional powers to simplify our calculations.

### Step 1: Evaluate Statement I
**Statement I:** \( (27)^{(1/3)} > (13)^{(1/2)} > (47)^{(1/6)} \)

1. **Find the LCM of the denominators:** The denominators are 3, 2, and 6. The LCM is 6.
2. **Convert each term:**
- \( (27)^{(1/3)} = (27)^{(2/6)} = 729^{(1/6)} \)
- \( (13)^{(1/2)} = (13)^{(3/6)} = 13^3 = 2197^{(1/6)} \)
- \( (47)^{(1/6)} = 47^{(1/6)} \)

3. **Compare the values:**
- We need to check if \( 729 > 2197 > 47 \).
- Clearly, \( 729 < 2197 \), so the first part of the statement is false.

Thus, **Statement I is false.**

### Step 2: Evaluate Statement II
**Statement II:** \( (23)^{(1/3)} < (49)^{(1/2)} < (52)^{(1/6)} \)

1. **Find the LCM of the denominators:** The denominators are 3, 2, and 6. The LCM is 6.
2. **Convert each term:**
- \( (23)^{(1/3)} = (23)^{(2/6)} = 23^2 = 529^{(1/6)} \)
- \( (49)^{(1/2)} = (49)^{(3/6)} = 49^3 = 117649^{(1/6)} \)
- \( (52)^{(1/6)} = 52^{(1/6)} \)

3. **Compare the values:**
- We need to check if \( 529 < 117649 < 52 \).
- Clearly, \( 529 < 117649 \) is true, but \( 117649 < 52 \) is false.

Thus, **Statement II is false.**

### Step 3: Evaluate Statement III
**Statement III:** \( (52)^{(1/6)} < (41)^{(1/3)} < (37)^{(1/2)} \)

1. **Find the LCM of the denominators:** The denominators are 6, 3, and 2. The LCM is 6.
2. **Convert each term:**
- \( (52)^{(1/6)} = 52^{(1/6)} \)
- \( (41)^{(1/3)} = (41)^{(2/6)} = 41^2 = 1681^{(1/6)} \)
- \( (37)^{(1/2)} = (37)^{(3/6)} = 37^3 = 50653^{(1/6)} \)

3. **Compare the values:**
- We need to check if \( 52 < 1681 < 50653 \).
- Clearly, \( 52 < 1681 \) and \( 1681 < 50653 \) are both true.

Thus, **Statement III is true.**

### Conclusion
The false statements are **Statement I** and **Statement II**.