Which one of the following is true?

Options

A. `sqrt5+sqrt3 gt sqrt 6+ sqrt 2`
B. `sqrt5+sqrt3 lt sqrt 6+ sqrt 2`
C. `sqrt5+sqrt3 = sqrt 6+ sqrt 2`
D. `(sqrt5+sqrt3)(sqrt 6+ sqrt 2) = 1`

Answer

To determine which of the following statements is true, we need to compare the values of \( \sqrt{5} + \sqrt{3} \) and \( \sqrt{6} + \sqrt{2} \). We can do this by squaring both expressions and comparing the results.

### Step-by-Step Solution:

1. **Square the first expression**:
\[ (\sqrt{5} + \sqrt{3})^2 = (\sqrt{5})^2 + (\sqrt{3})^2 + 2 \cdot \sqrt{5} \cdot \sqrt{3} \]
This simplifies to:
\[ 5 + 3 + 2\sqrt{15} = 8 + 2\sqrt{15} \]

2. **Square the second expression**:
\[ (\sqrt{6} + \sqrt{2})^2 = (\sqrt{6})^2 + (\sqrt{2})^2 + 2 \cdot \sqrt{6} \cdot \sqrt{2} \]
This simplifies to:
\[ 6 + 2 + 2\sqrt{12} = 8 + 2\sqrt{12} \]

3. **Compare the two squared results**:
We have:
\[ 8 + 2\sqrt{15} \quad \text{and} \quad 8 + 2\sqrt{12} \]
Since both expressions have the same constant term (8), we only need to compare the terms involving square roots:
\[ 2\sqrt{15} \quad \text{and} \quad 2\sqrt{12} \]

4. **Determine which square root is greater**:
We know that:
\[ \sqrt{15} > \sqrt{12} \]
Therefore:
\[ 2\sqrt{15} > 2\sqrt{12} \]

5. **Conclude the comparison**:
Since \( 8 + 2\sqrt{15} > 8 + 2\sqrt{12} \), we can conclude that:
\[ (\sqrt{5} + \sqrt{3})^2 > (\sqrt{6} + \sqrt{2})^2 \]
Taking the square root of both sides (since both sides are positive), we find:
\[ \sqrt{5} + \sqrt{3} > \sqrt{6} + \sqrt{2} \]

Thus, the correct answer is that \( \sqrt{5} + \sqrt{3} \) is greater than \( \sqrt{6} + \sqrt{2} \).

### Final Answer:
The correct option is option 1: \( \sqrt{5} + \sqrt{3} > \sqrt{6} + \sqrt{2} \).

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