Which of the following is TRUE ?
`I. root3(11) gt sqrt(7) gt root4(45)`
II. `sqrt(7) gt root3(11) gt root4(45)`
III `sqrt(7) gt root4(45) gt root3(11)`
IV. `root4(45) gt sqrt(7) gt root3(11)`

Options

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

Answer

To determine which of the given statements is true, we need to compare the values of \( \sqrt[3]{11} \), \( \sqrt{7} \), and \( \sqrt[4]{45} \).

### Step 1: Convert the roots to powers
We can express the roots in terms of powers:
- \( \sqrt[3]{11} = 11^{1/3} \)
- \( \sqrt{7} = 7^{1/2} \)
- \( \sqrt[4]{45} = 45^{1/4} \)

### Step 2: Find a common denominator for the powers
To compare these values, we will find a common denominator for the exponents. The denominators are 3, 2, and 4. The least common multiple (LCM) of these numbers is 12.

### Step 3: Rewrite each expression with the common denominator
Now we can rewrite each expression with a denominator of 12:
- \( 11^{1/3} = 11^{4/12} \)
- \( 7^{1/2} = 7^{6/12} \)
- \( 45^{1/4} = 45^{3/12} \)

### Step 4: Compare the values
Now we can compare:
- \( 11^{4/12} \)
- \( 7^{6/12} \)
- \( 45^{3/12} \)

### Step 5: Calculate the approximate values
To compare these values, we can calculate them:
1. \( 11^{4/12} = 11^{1/3} \approx 2.224 \)
2. \( 7^{6/12} = 7^{1/2} \approx 2.645 \)
3. \( 45^{3/12} = 45^{1/4} \approx 3.169 \)

### Step 6: Order the values
Now we can order these values:
- \( \sqrt[3]{11} \approx 2.224 \)
- \( \sqrt{7} \approx 2.645 \)
- \( \sqrt[4]{45} \approx 3.169 \)

From this, we can see:
\[ \sqrt[3]{11} < \sqrt{7} < \sqrt[4]{45} \]

### Conclusion
Thus, the correct order is:
\[ \sqrt[3]{11} < \sqrt{7} < \sqrt[4]{45} \]
This means the statement that is true is:
**III. \( \sqrt{7} > \sqrt[4]{45} > \sqrt[3]{11} \)**