If `p=sqrt(7)-sqrt(5)` and `q=sqrt(13)-sqrt(11)`, then

Options

A. `p gt q`
B. `p lt q`
C. `p=q`
D. none of these

Answer

To compare \( p = \sqrt{7} - \sqrt{5} \) and \( q = \sqrt{13} - \sqrt{11} \), we will calculate the approximate values of \( p \) and \( q \).

### Step 1: Calculate \( p \)

1. **Find \( \sqrt{7} \)**:
- We know that \( \sqrt{4} = 2 \) and \( \sqrt{9} = 3 \). Therefore, \( \sqrt{7} \) is between 2 and 3.
- A more precise approximation can be found by estimating: \( \sqrt{7} \approx 2.6457513110645906 \).

2. **Find \( \sqrt{5} \)**:
- We know that \( \sqrt{4} = 2 \) and \( \sqrt{9} = 3 \). Therefore, \( \sqrt{5} \) is between 2 and 3.
- A more precise approximation can be found by estimating: \( \sqrt{5} \approx 2.23606797749979 \).

3. **Calculate \( p \)**:
\[ p = \sqrt{7} - \sqrt{5} \approx 2.6457513110645906 - 2.23606797749979 \approx 0.4096833335648006 \]

### Step 2: Calculate \( q \)

1. **Find \( \sqrt{13} \)**:
- We know that \( \sqrt{9} = 3 \) and \( \sqrt{16} = 4 \). Therefore, \( \sqrt{13} \) is between 3 and 4.
- A more precise approximation can be found by estimating: \( \sqrt{13} \approx 3.605551275463989 \).

2. **Find \( \sqrt{11} \)**:
- We know that \( \sqrt{9} = 3 \) and \( \sqrt{16} = 4 \). Therefore, \( \sqrt{11} \) is between 3 and 4.
- A more precise approximation can be found by estimating: \( \sqrt{11} \approx 3.3166247903554 \).

3. **Calculate \( q \)**:
\[ q = \sqrt{13} - \sqrt{11} \approx 3.605551275463989 - 3.3166247903554 \approx 0.288926485108589 \]

### Step 3: Compare \( p \) and \( q \)

Now we have:
- \( p \approx 0.4096833335648006 \)
- \( q \approx 0.288926485108589 \)

Since \( p > q \), we conclude that:

\[ p > q \]

### Final Answer:
Thus, the relationship between \( p \) and \( q \) is \( p > q \).

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