An irrational number between `sqrt(2)` and `sqrt(3)` is .

Options

A. `(sqrt(2) + sqrt(3))`
B. `sqrt(2) xx sqrt(3)`
C. `5^(1//4)`
D. `6^(1//4)`

Answer

To find an irrational number between \( \sqrt{2} \) and \( \sqrt{3} \), we can follow these steps:

### Step 1: Identify the values of \( \sqrt{2} \) and \( \sqrt{3} \)
- We know that \( \sqrt{2} \) is approximately \( 1.414 \) and \( \sqrt{3} \) is approximately \( 1.732 \).

### Step 2: Add \( \sqrt{2} \) and \( \sqrt{3} \)
- Calculate \( \sqrt{2} + \sqrt{3} \):
\[ \sqrt{2} + \sqrt{3} \approx 1.414 + 1.732 = 3.146 \]

### Step 3: Divide the sum by 2
- To find a number between \( \sqrt{2} \) and \( \sqrt{3} \), we divide the sum by 2:
\[ \frac{\sqrt{2} + \sqrt{3}}{2} \approx \frac{3.146}{2} \approx 1.573 \]

### Step 4: Verify if the result is between \( \sqrt{2} \) and \( \sqrt{3} \)
- We need to check if \( 1.573 \) is between \( 1.414 \) and \( 1.732 \):
- Since \( 1.414 < 1.573 < 1.732 \), it confirms that \( \frac{\sqrt{2} + \sqrt{3}}{2} \) is indeed between \( \sqrt{2} \) and \( \sqrt{3} \).

### Step 5: Confirm that the result is irrational
- The sum of two irrational numbers \( \sqrt{2} \) and \( \sqrt{3} \) is also irrational, and dividing an irrational number by a rational number (2 in this case) results in an irrational number. Therefore, \( \frac{\sqrt{2} + \sqrt{3}}{2} \) is irrational.

### Final Answer
Thus, an irrational number between \( \sqrt{2} \) and \( \sqrt{3} \) is:
\[ \frac{\sqrt{2} + \sqrt{3}}{2} \]

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