We are asked to evaluate $f(g(h(x)))$ given the following functions:

$f(x) = x^4 + 6, \quad g(x) = x - 5, \quad h(x) = \sqrt{x}.$

Let's break this down step by step.

Step 1: Evaluate $h(x)$

$h(x) = \sqrt{x}.$

Step 2: Apply $g(x)$ to $h(x)$

We need to substitute $h(x) = \sqrt{x}$ into $g(x) = x - 5$. $g(h(x)) = g(\sqrt{x}) = \sqrt{x} - 5.$

Step 3: Apply $f(x)$ to $g(h(x))$

Now substitute $g(h(x)) = \sqrt{x} - 5$ into $f(x) = x^4 + 6$. $f(g(h(x))) = f(\sqrt{x} - 5) = (\sqrt{x} - 5)^4 + 6.$

Thus, the final expression for $f(g(h(x)))$ is: $f(g(h(x))) = (\sqrt{x} - 5)^4 + 6.$

Would you like further details on how to expand or simplify this result? Here are a few related questions you can consider:

1. What is the expanded form of $(\sqrt{x} - 5)^4$?
2. How do you compose functions in general?
3. What are some applications of composite functions in real-life problems?
4. How can you verify if two functions are inverses using compositions?
5. What happens if we change one of the functions in the composition?

Tip: When working with nested functions, start from the innermost function and work your way outward to avoid mistakes.