Which of the following statement (s) is /are true ?
I. `root(3)(7)gtroot(4)(11)`
II . `root(4)(4)gtroot(8)(12)`

Options

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

Answer

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

### Statement I: \( \sqrt[3]{7} > \sqrt[4]{11} \)

1. **Identify the roots**: We need to compare the cube root of 7 and the fourth root of 11.

2. **Find a common exponent**: The least common multiple (LCM) of the roots (3 and 4) is 12. We will raise both sides to the power of 12 to eliminate the roots:
\[ \left(\sqrt[3]{7}\right)^{12} \quad \text{and} \quad \left(\sqrt[4]{11}\right)^{12} \]

3. **Rewrite the expressions**:
\[ 7^{4} \quad \text{and} \quad 11^{3} \]

4. **Calculate the values**:
- \( 7^{4} = 2401 \)
- \( 11^{3} = 1331 \)

5. **Compare the results**:
\[ 2401 > 1331 \]
Therefore, statement I is true.

### Statement II: \( \sqrt[4]{4} > \sqrt[8]{12} \)

1. **Identify the roots**: We need to compare the fourth root of 4 and the eighth root of 12.

2. **Find a common exponent**: The least common multiple (LCM) of the roots (4 and 8) is 8. We will raise both sides to the power of 8:
\[ \left(\sqrt[4]{4}\right)^{8} \quad \text{and} \quad \left(\sqrt[8]{12}\right)^{8} \]

3. **Rewrite the expressions**:
\[ 4^{2} \quad \text{and} \quad 12^{1} \]

4. **Calculate the values**:
- \( 4^{2} = 16 \)
- \( 12^{1} = 12 \)

5. **Compare the results**:
\[ 16 > 12 \]
Therefore, statement II is also true.

### Conclusion
Both statements are true. Thus, the answer is:

**Both statements I and II are true.**