The number of values of x satisfying
`2^(log_5 16. log_4 x + log_(root(x)(2))5) + 5^x + x^((log_5 4)+5) + x^5 = 0`

Explanation

To solve the equation

\[ 2^{(\log_5 16) \cdot (\log_4 x) + \log_{\sqrt{x}} 5} + 5^x + x^{(\log_5 4) + 5} + x^5 = 0, \]

we will analyze each term step-by-step.

### Step 1: Simplify the logarithmic terms

We start by rewriting \(\log_5 16\) and \(\log_4 x\):

\[ \log_5 16 = \log_5 (4^2) = 2 \log_5 4. \]

Thus, we can rewrite the equation as:

\[ 2^{(2 \log_5 4) \cdot (\log_4 x) + \log_{\sqrt{x}} 5} + 5^x + x^{(\log_5 4) + 5} + x^5 = 0. \]

### Step 2: Rewrite \(\log_{\sqrt{x}} 5\)

Using the change of base formula, we can rewrite \(\log_{\sqrt{x}} 5\):

\[ \log_{\sqrt{x}} 5 = \frac{\log_5 5}{\log_5 \sqrt{x}} = \frac{1}{\frac{1}{2} \log_5 x} = \frac{2}{\log_5 x}. \]

### Step 3: Substitute back into the equation

Now substituting back, we have:

\[ 2^{(2 \log_5 4) \cdot (\log_4 x) + \frac{2}{\log_5 x}} + 5^x + x^{(\log_5 4) + 5} + x^5 = 0. \]

### Step 4: Analyze the terms

1. **First Term**: \(2^{(2 \log_5 4) \cdot (\log_4 x) + \frac{2}{\log_5 x}}\) is always positive for any \(x > 0\).
2. **Second Term**: \(5^x\) is also always positive for any real \(x\).
3. **Third Term**: \(x^{(\log_5 4) + 5}\) is positive for \(x > 0\).
4. **Fourth Term**: \(x^5\) is positive for \(x > 0\).

### Step 5: Conclude about the equation

Since all terms are positive for \(x > 0\), the left-hand side of the equation cannot equal zero for any positive \(x\).

### Step 6: Consider negative values of \(x\)

For \(x < 0\), the logarithmic terms become undefined, as logarithms are not defined for negative numbers in the real number system.

### Conclusion

Thus, there are no values of \(x\) that satisfy the given equation. Therefore, the number of values of \(x\) satisfying the equation is:

\[ \boxed{0}. \]

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