The square root of `(3-sqrt5)` is

Options

A. `(sqrt3-5^(1/4))`
B. `1/2(sqrt5-sqrt3)`
C. `1/2(sqrt5-1)`
D. `1/(sqrt2)(sqrt5-1)`

Answer

To find the square root of the expression \(3 - \sqrt{5}\), we can follow these steps:

### Step 1: Rewrite the expression
We start with the expression:
\[ \sqrt{3 - \sqrt{5}} \]

### Step 2: Rationalize the expression
To simplify the square root, we can express \(3 - \sqrt{5}\) in a form that is easier to work with. We can multiply the numerator and denominator by a suitable expression to eliminate the square root in the denominator. Here, we will multiply by \(2\) to help rationalize it:
\[ \sqrt{3 - \sqrt{5}} = \sqrt{\frac{(3 - \sqrt{5}) \cdot 2}{2}} = \frac{\sqrt{2(3 - \sqrt{5})}}{2} \]

### Step 3: Expand the numerator
Now, we can expand the numerator:
\[ 2(3 - \sqrt{5}) = 6 - 2\sqrt{5} \]
So, we have:
\[ \sqrt{3 - \sqrt{5}} = \frac{\sqrt{6 - 2\sqrt{5}}}{2} \]

### Step 4: Factor the expression under the square root
Next, we can factor \(6 - 2\sqrt{5}\) into a perfect square. We can express it as:
\[ 6 - 2\sqrt{5} = (\sqrt{a} - \sqrt{b})^2 \]
where \(a + b = 6\) and \(2\sqrt{ab} = 2\sqrt{5}\). Solving these equations, we find \(a = 5\) and \(b = 1\). Thus:
\[ 6 - 2\sqrt{5} = (\sqrt{5} - 1)^2 \]

### Step 5: Take the square root
Now, we can take the square root:
\[ \sqrt{6 - 2\sqrt{5}} = \sqrt{(\sqrt{5} - 1)^2} = \sqrt{5} - 1 \]

### Step 6: Final expression
Putting it all together, we have:
\[ \sqrt{3 - \sqrt{5}} = \frac{\sqrt{5} - 1}{2} \]

### Conclusion
Thus, the square root of \(3 - \sqrt{5}\) is:
\[ \frac{\sqrt{5} - 1}{2} \]

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