Which of the numbers given below is rational ?
A)`root(6)(49)`
B)`root(6)(81)`
C)`root(6)(64)`
D)`root(6)(36)`

Options

A. `root(6)(49)`
B. `root(6)(81)`
C. `root(6)(64)`
D. `root(6)(36)`

Answer

To determine which of the given numbers is rational, we need to evaluate each option and check if it can be expressed as a fraction \( \frac{P}{Q} \) where \( P \) and \( Q \) are integers and \( Q \neq 0 \).

### Step-by-Step Solution:

1. **Understanding Rational Numbers**:
A rational number can be expressed in the form \( \frac{P}{Q} \), where \( P \) and \( Q \) are integers and \( Q \neq 0 \).

2. **Evaluating Option A: \( \sqrt[6]{49} \)**:
- \( 49 = 7^2 \)
- Therefore, \( \sqrt[6]{49} = (7^2)^{1/6} = 7^{2/6} = 7^{1/3} \).
- \( 7^{1/3} \) is not a rational number because it cannot be expressed as a fraction of integers.

**Hint**: Check if the base can be expressed as a power of a rational number.

3. **Evaluating Option B: \( \sqrt[6]{81} \)**:
- \( 81 = 3^4 \)
- Therefore, \( \sqrt[6]{81} = (3^4)^{1/6} = 3^{4/6} = 3^{2/3} \).
- \( 3^{2/3} \) is also not a rational number.

**Hint**: Simplify the expression and see if the exponent results in a rational number.

4. **Evaluating Option C: \( \sqrt[6]{64} \)**:
- \( 64 = 2^6 \)
- Therefore, \( \sqrt[6]{64} = (2^6)^{1/6} = 2^{6/6} = 2^1 = 2 \).
- \( 2 \) is a rational number since it can be expressed as \( \frac{2}{1} \).

**Hint**: Look for perfect powers that simplify to integers.

5. **Evaluating Option D: \( \sqrt[6]{36} \)**:
- \( 36 = 6^2 \) or \( 36 = 2^2 \times 3^2 \)
- Therefore, \( \sqrt[6]{36} = (6^2)^{1/6} = 6^{2/6} = 6^{1/3} \).
- \( 6^{1/3} \) is not a rational number.

**Hint**: Factor the number and check if the resulting exponent leads to a rational number.

### Conclusion:
The only rational number among the options is **Option C: \( \sqrt[6]{64} = 2 \)**.