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 \( 5^{\log_{1/5}(1/2)} + \frac{\log_{\sqrt{2}}(4)}{(\sqrt{7} + \sqrt{3})} + \log_{1/2}\left(\frac{1}{10 + 2\sqrt{21}}\right) \), we can break it down into three parts and simplify each one step by step.

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

Using the change of base formula, we can rewrite \( \log_{1/5}(1/2) \) as:
\[ \log_{1/5}(1/2) = \frac{\log(1/2)}{\log(1/5)} \]
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