The inequality you're given is:  
\[ x^{-8x} + 16 > 0 \]

This appears to be a quadratic inequality, but I believe there might be a small mistake in how it's written. It looks like you might have meant to write it as:

\[
x^2 - 8x + 16 > 0
\]

If so, let's break this inequality down:

### 1. Solve the quadratic equation:
First, let's consider the corresponding equation:

\[
x^2 - 8x + 16 = 0
\]

This is a perfect square trinomial:
\[
(x - 4)^2 = 0
\]
So, the only solution to this quadratic equation is \( x = 4 \).

### 2. Solve the inequality:
We need to determine where the expression \( x^2 - 8x + 16 > 0 \). Since the quadratic expression can be rewritten as \( (x - 4)^2 \), this is a square of a binomial. Therefore, \( (x - 4)^2 \geq 0 \) for all values of \(x\), and it equals zero only when \(x = 4\).

Thus, the inequality \( x^2 - 8x + 16 > 0 \) holds for all values of \( x \) **except** \( x = 4 \), where the expression is equal to zero.

### Final Solution:
\[
x \in (-\infty, 4) \cup (4, \infty)
\]

This means the inequality holds for all values of \( x \) except at \( x = 4 \).

---

Do you want more details or have any questions on this?

Here are some related questions to explore further:

1. How do you solve quadratic inequalities with non-perfect square trinomials?
2. How do you find the roots of a general quadratic equation?
3. What is the geometric interpretation of quadratic inequalities?
4. How do you solve inequalities involving squares like \( (x - a)^2 > 0 \)?
5. How does the discriminant of a quadratic equation affect the inequality solution?

**Tip:** For quadratic inequalities, always start by solving the corresponding equation to find the critical points.

Learn how to solve the quadratic inequality x^2 - 8x + 16 > 0 with a clear step-by-step guide. Explore concepts like quadratic equations, perfect square trinomials, and solving inequalities. This solution is perfect for students in Grades 9-12.

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