To determine the simplest form of the given expression [tex]\(\frac{3-\sqrt{5}}{\sqrt{5}}\)[/tex], we'll follow these steps:
1. Original Expression:
[tex]\[
\frac{3 - \sqrt{5}}{\sqrt{5}}
\][/tex]
2. Rationalizing the Denominator: We multiply the numerator and the denominator by [tex]\(\sqrt{5}\)[/tex] to rationalize it.
[tex]\[
\frac{(3 - \sqrt{5}) \cdot \sqrt{5}}{\sqrt{5} \cdot \sqrt{5}} = \frac{(3 \sqrt{5} - 5)}{5}
\][/tex]
3. Simplified Expression: The rationalized form of the expression is:
[tex]\[
\frac{3 \sqrt{5} - 5}{5}
\][/tex]
Now, we compare this result with the given options:
- Option 1: [tex]\(\frac{3 \sqrt{5} - 5}{5}\)[/tex]
- Option 2: [tex]\(\frac{3 \sqrt{3} - \sqrt{15}}{5}\)[/tex]
- Option 3: [tex]\(\frac{5 \sqrt{3} - 5}{5}\)[/tex]
- Option 4: [tex]\(\frac{14 - 6 \sqrt{5}}{5}\)[/tex]
From our simplified expression, we see that it matches exactly with Option 1:
[tex]\[
\frac{3 \sqrt{5} - 5}{5}
\][/tex]
Thus, the simplest form of the expression [tex]\(\frac{3-\sqrt{5}}{\sqrt{5}}\)[/tex] from the given options is:
[tex]\[
\frac{3 \sqrt{5} - 5}{5}
\][/tex]
Therefore, the correct answer is:
[tex]\[
\boxed{\frac{3 \sqrt{5} - 5}{5}}
\][/tex]
