The value of `5^((log)_(1/5)(1/2))+(log)_(sqrt(2))4/(sqrt(7)+sqrt(3))+(log)_(1/2)1/(10+2sqrt(21))` is.........

Explanation

To solve the expression \(5^{\log_{1/5}(1/2)} + \frac{\log_{\sqrt{2}} 4}{\sqrt{7} + \sqrt{3}} + \log_{1/2} \frac{1}{10 + 2\sqrt{21}}\), we will break it down into parts and simplify each part step by step.

### Step 1: Simplify \(5^{\log_{1/5}(1/2)}\)

Using the change of base formula:
\[ \log_{1/5}(1/2) = \frac{\log_{5}(1/2)}{\log_{5}(1/5)} = \log_{5}(1/2) \cdot (-1) = -\log_{5}(1/2) \]
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