1. **State the problem:** We are given three functions: $$f(x) = x^4 + 8$$ $$g(x) = x - 6$$ $$h(x) = \sqrt{x}$$ We need to find the composition $$f(g(h(x)))$$. 2. **Recall the composition rule:** For functions $f$, $g$, and $h$, the composition $f(g(h(x)))$ means we first apply $h$ to $x$, then apply $g$ to the result, and finally apply $f$ to that result. 3. **Calculate $h(x)$:** $$h(x) = \sqrt{x}$$ 4. **Calculate $g(h(x))$:** Substitute $h(x)$ into $g$: $$g(h(x)) = g(\sqrt{x}) = \sqrt{x} - 6$$ 5. **Calculate $f(g(h(x)))$:** Substitute $g(h(x))$ into $f$: $$f(g(h(x))) = f(\sqrt{x} - 6) = (\sqrt{x} - 6)^4 + 8$$ 6. **Final answer:** $$\boxed{f(g(h(x))) = (\sqrt{x} - 6)^4 + 8}$$ This expression represents the composition of the three functions as requested.