The value of `log_10(sqrt(3-sqrt(5))+sqrt(3+sqrt(5)))` is

Options

A. `1//2`
B. `1//4`
C. `3//2`
D. `3//4`

Explanation

To find the value of \( \log_{10}(\sqrt{3 - \sqrt{5}} + \sqrt{3 + \sqrt{5}}) \), we can follow these steps:

### Step 1: Simplify the Expression
We start with the expression inside the logarithm:
\[ \sqrt{3 - \sqrt{5}} + \sqrt{3 + \sqrt{5}} \]

### Step 2: Rationalize the Terms
We can multiply and divide by \( \sqrt{2} \):
\[ \sqrt{3 - \sqrt{5}} + \sqrt{3 + \sqrt{5}} = \frac{1}{\sqrt{2}} \left( \sqrt{2(3 - \sqrt{5})} + \sqrt{2(3 + \sqrt{5})} \right) \]

### Step 3: Expand the Terms
Now we can simplify the terms under the square roots:
\[ \sqrt{2(3 - \sqrt{5})} = \sqrt{6 - 2\sqrt{5}}, \quad \sqrt{2(3 + \sqrt{5})} = \sqrt{6 + 2\sqrt{5}} \]

### Step 4: Recognize Perfect Squares
Notice that:
\[ 6 - 2\sqrt{5} = (\sqrt{5} - 1)^2 \quad \text{and} \quad 6 + 2\sqrt{5} = (\sqrt{5} + 1)^2 \]
Thus, we can write:
\[ \sqrt{6 - 2\sqrt{5}} = \sqrt{(\sqrt{5} - 1)^2} = \sqrt{5} - 1 \]
\[ \sqrt{6 + 2\sqrt{5}} = \sqrt{(\sqrt{5} + 1)^2} = \sqrt{5} + 1 \]

### Step 5: Combine the Terms
Now, substituting back:
\[ \sqrt{3 - \sqrt{5}} + \sqrt{3 + \sqrt{5}} = \frac{1}{\sqrt{2}} \left( (\sqrt{5} - 1) + (\sqrt{5} + 1) \right) = \frac{1}{\sqrt{2}} (2\sqrt{5}) = \sqrt{10} \]

### Step 6: Calculate the Logarithm
Now we can find the logarithm:
\[ \log_{10}(\sqrt{10}) = \log_{10}(10^{1/2}) = \frac{1}{2} \log_{10}(10) = \frac{1}{2} \cdot 1 = \frac{1}{2} \]

### Final Answer
Thus, the value of \( \log_{10}(\sqrt{3 - \sqrt{5}} + \sqrt{3 + \sqrt{5}}) \) is:
\[ \frac{1}{2} \]

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