Which of the following statements(s) / are TRUE ?
I. `33^(3) gt 3^(33)`
II. `333 gt (3^(3))^(3)`

Options

A. Only I
B. Only II
C. Both I and II
D. Neither I nor II

Answer

To determine the truth of the statements given, we need to analyze each one step by step.

### Statement I: \( 33^3 > 3^{33} \)

1. **Rewrite the bases**: We can express \( 33 \) in terms of \( 3 \):
\[ 33 = 3 \times 11 \]
Therefore, we can rewrite \( 33^3 \) as:
\[ (3 \times 11)^3 = 3^3 \times 11^3 \]

2. **Compare the expressions**: Now we have:
\[ 3^3 \times 11^3 \quad \text{and} \quad 3^{33} \]
We can express \( 3^{33} \) as:
\[ 3^{33} = 3^3 \times 3^{30} \]

3. **Simplify the comparison**: We can divide both sides by \( 3^3 \) (since \( 3^3 > 0 \)):
\[ 11^3 > 3^{30} \]

4. **Calculate the values**:
- \( 11^3 = 1331 \)
- \( 3^{30} \) is a much larger number, as \( 3^{10} = 59049 \) and \( 3^{30} = (3^{10})^3 = 59049^3 \).

5. **Conclusion for Statement I**: Since \( 3^{30} \) is much larger than \( 1331 \), we conclude:
\[ 33^3 < 3^{33} \]
Thus, Statement I is **FALSE**.

### Statement II: \( 33 > (3^3)^3 \)

1. **Calculate \( (3^3)^3 \)**:
\[ (3^3)^3 = 3^{3 \times 3} = 3^9 \]

2. **Calculate \( 3^9 \)**:
- \( 3^9 = 19683 \)

3. **Compare the values**:
- We need to compare \( 33 \) and \( 19683 \):
\[ 33 < 19683 \]

4. **Conclusion for Statement II**: Thus, Statement II is also **FALSE**.

### Final Conclusion:
Both statements are false, so the correct answer is that neither statement is true.