Rationalizing a denominator using the conjugate method to eliminate radicals.

To rationalize the denominator of the fraction $\frac{1}{\sqrt{5} + \sqrt{3}}$, we follow these steps:

Step 1: Identify the conjugate.
The denominator is $\sqrt{5} + \sqrt{3}$. To remove the square roots from the bottom, we need to multiply by its conjugate. The conjugate is found by changing the sign in the middle.
Conjugate: $\sqrt{5} - \sqrt{3}$

Step 2: Multiply the numerator and denominator by the conjugate.
$$ \frac{1}{\sqrt{5} + \sqrt{3}} \times \frac{\sqrt{5} - \sqrt{3}}{\sqrt{5} - \sqrt{3}} $$

Step 3: Simplify the numerator.
Multiply the top terms:
$$ 1 \times (\sqrt{5} - \sqrt{3}) = \sqrt{5} - \sqrt{3} $$

Step 4: Simplify the denominator.
Multiply the bottom terms using the difference of squares formula $(a+b)(a-b) = a^2 - b^2$:
$$ (\sqrt{5} + \sqrt{3})(\sqrt{5} - \sqrt{3}) = (\sqrt{5})^2 - (\sqrt{3})^2 $$
$$ = 5 - 3 $$
$$ = 2 $$

Step 5: Write the final fraction.
Combine the simplified numerator and denominator:
$$ \frac{\sqrt{5} - \sqrt{3}}{2} $$

Final Answer:
\frac{\sqrt{5} - \sqrt{3}}{2}