Which of the following relation (s) is /are true ?
I. `3^(33) gt 33^(3)`
II. `33^(3) gt 333`
III. `3^(33) gt 333`

Options

A. Only I and II
B. Only II and III
C. Only II
D. All I, II and III

Answer

To determine which of the given relations are true, we will evaluate each statement one by one.

### Given Relations:
1. **I. \(3^{33} > 33^{3}\)**
2. **II. \(33^{3} > 333\)**
3. **III. \(3^{33} > 333\)**

### Step-by-Step Solution:

#### Step 1: Evaluate \(3^{33}\) and \(33^{3}\)
- We can express \(33\) as \(3 \times 11\).
- Therefore, \(33^{3} = (3 \times 11)^{3} = 3^{3} \times 11^{3} = 27 \times 1331 = 35937\).
- Now, we need to compare \(3^{33}\) with \(35937\).

Calculating \(3^{33}\):
- \(3^{33}\) is a very large number. We can estimate it using logarithms or direct calculation, but we know that \(3^{10} = 59049\) and \(3^{20} = (3^{10})^{2} = 59049^{2}\), which is much larger than \(35937\).
- Thus, \(3^{33} > 33^{3}\).

**Conclusion for I:** True

#### Step 2: Evaluate \(33^{3}\) and \(333\)
- We already calculated \(33^{3} = 35937\).
- Now, \(333\) is simply \(333\).
- Clearly, \(35937 > 333\).

**Conclusion for II:** True

#### Step 3: Evaluate \(3^{33}\) and \(333\)
- We already know \(3^{33}\) is a very large number, and we can compare it directly with \(333\).
- Since \(3^{10} = 59049\) and \(3^{33}\) is significantly larger than \(59049\), it is also much larger than \(333\).

**Conclusion for III:** True

### Final Results:
- **I. True**
- **II. True**
- **III. True**

### Summary:
All three relations are true.