Which of the following is TRUE ?
I. `(1)/(root3(12)) gt (1)/(root4(29)) gt (1)/(sqrt(5))`
II. `(1)/(root4(29)) gt (1)/(root3(12)) gt (1)/(sqrt(5))`
`III. (1)/(sqrt(5)) gt (1)/(root3(12)) gt (1)/(root4(29))`
`IV. (1)/(sqrt(5)) gt (1)/(root4(29)) gt (1)/(root3(12))`

Options

A. Only I
B. Only II
C. Only III
D. Only IV

Answer

To solve the problem, we need to compare the values of the following expressions:

1. \( \frac{1}{\sqrt[3]{12}} \)
2. \( \frac{1}{\sqrt[4]{29}} \)
3. \( \frac{1}{\sqrt{5}} \)

We will determine which of the four statements is true by calculating the approximate values of these expressions.

### Step 1: Calculate \( \sqrt[3]{12} \)

To find \( \sqrt[3]{12} \):
- We can express \( 12 \) as \( 2^2 \times 3 \).
- Therefore, \( \sqrt[3]{12} = \sqrt[3]{2^2 \times 3} = 2^{2/3} \times 3^{1/3} \).

Calculating \( \sqrt[3]{12} \) approximately:
- \( \sqrt[3]{2} \approx 1.26 \)
- \( \sqrt[3]{3} \approx 1.44 \)

Thus,
\[ \sqrt[3]{12} \approx 1.26^{2} \times 1.44 \approx 1.5874 \]

### Step 2: Calculate \( \sqrt[4]{29} \)

To find \( \sqrt[4]{29} \):
- We can express \( 29 \) as it is since it is a prime number.
- Therefore, \( \sqrt[4]{29} \approx 29^{0.25} \).

Calculating \( \sqrt[4]{29} \) approximately:
- \( \sqrt[4]{29} \approx 2.34 \)

### Step 3: Calculate \( \sqrt{5} \)

To find \( \sqrt{5} \):
- \( \sqrt{5} \approx 2.236 \).

### Step 4: Calculate the reciprocals

Now we calculate the reciprocals of these values:

1. \( \frac{1}{\sqrt[3]{12}} \approx \frac{1}{1.5874} \approx 0.629 \)
2. \( \frac{1}{\sqrt[4]{29}} \approx \frac{1}{2.34} \approx 0.427 \)
3. \( \frac{1}{\sqrt{5}} \approx \frac{1}{2.236} \approx 0.447 \)

### Step 5: Compare the values

Now we compare the values:
- \( \frac{1}{\sqrt[3]{12}} \approx 0.629 \)
- \( \frac{1}{\sqrt[4]{29}} \approx 0.427 \)
- \( \frac{1}{\sqrt{5}} \approx 0.447 \)

From the comparisons:
- \( 0.629 > 0.447 \)
- \( 0.447 > 0.427 \)

Thus, we can conclude:
\[ \frac{1}{\sqrt[3]{12}} > \frac{1}{\sqrt{5}} > \frac{1}{\sqrt[4]{29}} \]

### Conclusion

The correct statement is:
- III. \( \frac{1}{\sqrt{5}} > \frac{1}{\sqrt[4]{29}} > \frac{1}{\sqrt[3]{12}} \)