Simplify `:""^(3)sqrt(5^((1)/(log_(7)5))+(1)/(sqrt(-log_(10)(0.1))))`

Answer

To simplify the expression \( \sqrt[3]{5^{\frac{1}{\log_{7}5}} + \frac{1}{\sqrt{-\log_{10}(0.1)}}} \), we will follow these steps:

### Step 1: Simplify \( \log_{10}(0.1) \)
We know that \( 0.1 = \frac{1}{10} = 10^{-1} \). Therefore, we can use the logarithmic identity:
\[ \log_{10}(0.1) = \log_{10}(10^{-1}) = -1 \]

### Step 2: Substitute into the expression
Now substituting this back into our expression, we have:
\[ -\log_{10}(0.1) = -(-1) = 1 \]
Thus, we can rewrite the expression:
\[ \sqrt[3]{5^{\frac{1}{\log_{7}5}} + \frac{1}{\sqrt{1}}} \]
This simplifies to:
\[ \sqrt[3]{5^{\frac{1}{\log_{7}5}} + 1} \]

### Step 3: Simplify \( 5^{\frac{1}{\log_{7}5}} \)
Using the change of base formula for logarithms, we have:
\[ \frac{1}{\log_{7}5} = \log_{5}7 \]
So, we can rewrite \( 5^{\frac{1}{\log_{7}5}} \) as:
\[ 5^{\log_{5}7} = 7 \]

### Step 4: Substitute back into the expression
Now substituting this back into the expression, we have:
\[ \sqrt[3]{7 + 1} = \sqrt[3]{8} \]

### Step 5: Final simplification
The cube root of 8 is:
\[ \sqrt[3]{8} = 2 \]

Thus, the simplified expression is:
\[ \boxed{2} \]