Which of the following satements (s) is/are true ?
I. `2^(300) gt 3^(200)`
II. `6^(100) gt 2^(300)`

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 will analyze each statement step by step.

### Statement I: \(2^{300} > 3^{200}\)

1. **Rewrite the powers**: We can express both sides of the inequality in terms of a common exponent. We can rewrite \(2^{300}\) and \(3^{200}\) as follows:
\[ 2^{300} = (2^3)^{100} = 8^{100} \]
\[ 3^{200} = (3^2)^{100} = 9^{100} \]

2. **Compare the bases**: Now we have:
\[ 8^{100} > 9^{100} \]
Since \(8 < 9\), it follows that:
\[ 8^{100} < 9^{100} \]

3. **Conclusion for Statement I**: Therefore, the statement \(2^{300} > 3^{200}\) is **false**.

### Statement II: \(6^{100} > 2^{300}\)

1. **Rewrite the powers**: We can express \(6^{100}\) in terms of its prime factors:
\[ 6^{100} = (2 \cdot 3)^{100} = 2^{100} \cdot 3^{100} \]

2. **Rewrite \(2^{300}\)**: We already know from the previous analysis that:
\[ 2^{300} = (2^3)^{100} = 8^{100} \]

3. **Compare the two expressions**: Now we need to compare \(2^{100} \cdot 3^{100}\) with \(8^{100}\):
\[ 2^{100} \cdot 3^{100} \text{ vs } 8^{100} \]
We can rewrite \(8^{100}\) as:
\[ 8^{100} = (2^3)^{100} = 2^{300} \]

4. **Simplifying the comparison**: Therefore, we need to check if:
\[ 2^{100} \cdot 3^{100} > 2^{300} \]
Dividing both sides by \(2^{100}\) (since \(2^{100} > 0\)):
\[ 3^{100} > 2^{200} \]

5. **Comparing the bases**: We can rewrite \(2^{200}\) as:
\[ 2^{200} = (2^2)^{100} = 4^{100} \]
Now we need to compare \(3^{100}\) with \(4^{100}\):
\[ 3^{100} < 4^{100} \]
Since \(3 < 4\), it follows that:
\[ 3^{100} < 4^{100} \]

6. **Conclusion for Statement II**: Therefore, the statement \(6^{100} > 2^{300}\) is also **false**.

### Final Conclusion:
Both statements I and II are false. Thus, the answer is that neither statement is true.

---