Which of the following statement is true ?
`I. 3^(2^(3)) gt(3^(2))^(3)`
II. `2^(3^(2)) gt 2^(2^(3))`

Options

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

Answer

To determine which of the statements is true, we will analyze each statement step by step.

### Statement I: \( 3^{2^3} > (3^2)^3 \)

1. **Calculate \( 2^3 \)**:
\[ 2^3 = 8 \]
So, the left side becomes \( 3^8 \).

2. **Calculate \( (3^2)^3 \)**:
\[ (3^2)^3 = 3^{2 \cdot 3} = 3^6 \]
So, the right side is \( 3^6 \).

3. **Compare \( 3^8 \) and \( 3^6 \)**:
Since the base (3) is the same and the exponent 8 is greater than 6, we have:
\[ 3^8 > 3^6 \]
Therefore, Statement I is true.

### Statement II: \( 2^{3^2} > 2^{2^3} \)

1. **Calculate \( 3^2 \)**:
\[ 3^2 = 9 \]
So, the left side becomes \( 2^9 \).

2. **Calculate \( 2^3 \)**:
\[ 2^3 = 8 \]
So, the right side becomes \( 2^8 \).

3. **Compare \( 2^9 \) and \( 2^8 \)**:
Since the base (2) is the same and the exponent 9 is greater than 8, we have:
\[ 2^9 > 2^8 \]
Therefore, Statement II is also true.

### Conclusion:
Both statements are true.

### Final Answer:
Both statements I and II are true.

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