What is the value of `(sqrt5-sqrt3)div(sqrt5+sqrt3)`?

Options

A. `4-sqrt15`
B. `6-2sqrt15`
C. `8-2sqrt15`
D. `4+sqrt15`

Answer

To find the value of \((\sqrt{5} - \sqrt{3}) \div (\sqrt{5} + \sqrt{3})\), we can follow these steps:

### Step-by-Step Solution:

1. **Write the expression**:
\[ \frac{\sqrt{5} - \sqrt{3}}{\sqrt{5} + \sqrt{3}} \]

2. **Multiply the numerator and denominator by the conjugate of the denominator**:
\[ \frac{(\sqrt{5} - \sqrt{3})(\sqrt{5} - \sqrt{3})}{(\sqrt{5} + \sqrt{3})(\sqrt{5} - \sqrt{3})} \]

3. **Simplify the denominator using the difference of squares formula**:
\[ (\sqrt{5})^2 - (\sqrt{3})^2 = 5 - 3 = 2 \]

4. **Expand the numerator**:
\[ (\sqrt{5} - \sqrt{3})^2 = (\sqrt{5})^2 - 2(\sqrt{5})(\sqrt{3}) + (\sqrt{3})^2 = 5 - 2\sqrt{15} + 3 = 8 - 2\sqrt{15} \]

5. **Combine the results**:
\[ \frac{8 - 2\sqrt{15}}{2} \]

6. **Simplify the fraction**:
\[ \frac{8}{2} - \frac{2\sqrt{15}}{2} = 4 - \sqrt{15} \]

### Final Answer:
The value of \((\sqrt{5} - \sqrt{3}) \div (\sqrt{5} + \sqrt{3})\) is:
\[ 4 - \sqrt{15} \]