If $x=3-\sqrt{5}$, then $\frac{\sqrt{x}}{\sqrt{2}+\sqrt{3x-2}}=$

Question

Accepted Answer

To solve the problem and justify why Option 4 is correct, we need to evaluate the expression $\frac{\sqrt{x}}{\sqrt{2}+\sqrt{3x-2}}$ given that $x = 3 - \sqrt{5}$.

**Step-by-step Explanation:**

1. **Substitute $x$ into the expression:**
   - We have $x = 3 - \sqrt{5}$.
   - Therefore, $\sqrt{x} = \sqrt{3 - \sqrt{5}}$.

2. **Simplify the denominator:**
   - Calculate $3x - 2$:
     $$
     3x - 2 = 3(3 - \sqrt{5}) - 2 = 9 - 3\sqrt{5} - 2 = 7 - 3\sqrt{5}
     $$
   - The denominator becomes $\sqrt{2} + \sqrt{7 - 3\sqrt{5}}$.

3. **Rationalize the denominator:**
   - To simplify the expression, we rationalize the denominator by multiplying the numerator and the denominator by the conjugate of the denominator:
     $$
     \frac{\sqrt{3 - \sqrt{5}}}{\sqrt{2} + \sqrt{7 - 3\sqrt{5}}} \times \frac{\sqrt{2} - \sqrt{7 - 3\sqrt{5}}}{\sqrt{2} - \sqrt{7 - 3\sqrt{5}}}
     $$
   - The denominator becomes:
     $$
     (\sqrt{2} + \sqrt{7 - 3\sqrt{5}})(\sqrt{2} - \sqrt{7 - 3\sqrt{5}}) = 2 - (7 - 3\sqrt{5}) = 2 - 7 + 3\sqrt{5} = -5 + 3\sqrt{5}
     $$

4. **Evaluate the expression:**
   - The expression simplifies to:
     $$
     \frac{\sqrt{3 - \sqrt{5}}(\sqrt{2} - \sqrt{7 - 3\sqrt{5}})}{-5 + 3\sqrt{5}}
     $$
   - Further simplification shows that the expression evaluates to $\frac{1}{\sqrt{5}}$.

**Why Option 4 is Correct:**

- The expression simplifies to $\frac{1}{\sqrt{5}}$, which matches Option 4.

**Why Other Options are Incorrect:**

1. **Option 1 (5):**
   - The expression does not simplify to 5. The calculations show that the denominator and numerator do not result in a value of 5.

2. **Option 2 ($\sqrt{5}$):**
   - The expression does not simplify to $\sqrt{5}$. The rationalization and simplification process clearly leads to $\frac{1}{\sqrt{5}}$.

3. **Option 3 ($\frac{1}{5}$):**
   - The expression does not simplify to $\frac{1}{5}$. The calculations show that the denominator involves a square root, leading to $\frac{1}{\sqrt{5}}$.

Thus, the correct answer is indeed Option 4, $\frac{1}{\sqrt{5}}$.