Which of the following irrational ?

Options

A. `√((4)/(9))`
B. `(√(12))/(√(3))`
C. `√(7)`
D. `√(81)`

Answer

To determine which of the given options is irrational, we will analyze each option step by step.

### Step 1: Analyze Option A: \( \sqrt{\frac{4}{9}} \)

1. **Calculate the square root:**
\[ \sqrt{\frac{4}{9}} = \frac{\sqrt{4}}{\sqrt{9}} = \frac{2}{3} \]

2. **Determine if it is rational:**
- \( \frac{2}{3} \) can be expressed in the form \( \frac{p}{q} \) where \( p = 2 \) and \( q = 3 \) (and \( q \neq 0 \)).
- Therefore, \( \frac{2}{3} \) is a rational number.

### Step 2: Analyze Option B: \( \frac{\sqrt{12}}{\sqrt{3}} \)

1. **Simplify \( \sqrt{12} \):**
\[ \sqrt{12} = \sqrt{4 \times 3} = \sqrt{4} \times \sqrt{3} = 2\sqrt{3} \]

2. **Substitute back into the expression:**
\[ \frac{\sqrt{12}}{\sqrt{3}} = \frac{2\sqrt{3}}{\sqrt{3}} = 2 \]

3. **Determine if it is rational:**
- The number \( 2 \) can also be expressed as \( \frac{2}{1} \).
- Therefore, \( 2 \) is a rational number.

### Step 3: Analyze Option C: \( \sqrt{7} \)

1. **Determine if it is rational:**
- The number \( \sqrt{7} \) cannot be expressed in the form \( \frac{p}{q} \) where \( p \) and \( q \) are integers.
- \( \sqrt{7} \) is an irrational number.

### Step 4: Analyze Option D: \( \sqrt{81} \)

1. **Calculate the square root:**
\[ \sqrt{81} = 9 \]

2. **Determine if it is rational:**
- The number \( 9 \) can be expressed as \( \frac{9}{1} \).
- Therefore, \( 9 \) is a rational number.

### Conclusion

After analyzing all options, we find that the only irrational number among the options is:

**Option C: \( \sqrt{7} \)**

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