Let's break down the compositions of functions based on the given definitions of $f(x) = 8^x$ and $g(x) = x^8$.

1. $(f \circ f)(x) = f(f(x))$

• We know that $f(x) = 8^x$, so $f(f(x)) = f(8^x)$.
• Applying $f$ again to $8^x$, we get: $f(8^x) = 8^{8^x}.$ Thus, $(f \circ f)(x) = 8^{8^x}$.

2. $(f \circ g)(x) = f(g(x))$

• Since $g(x) = x^8$, we substitute $g(x)$ into $f(x)$: $f(g(x)) = f(x^8) = 8^{x^8}.$ Thus, $(f \circ g)(x) = 8^{x^8}$.

3. $(g \circ f)(x) = g(f(x))$

• Since $f(x) = 8^x$, we substitute $f(x)$ into $g(x)$: $g(f(x)) = g(8^x) = (8^x)^8 = 8^{8x}.$ Thus, $(g \circ f)(x) = 8^{8x}$.

4. $(g \circ g)(x) = g(g(x))$

• We know that $g(x) = x^8$, so $g(g(x)) = g(x^8) = (x^8)^8 = x^{64}$. Thus, $(g \circ g)(x) = x^{64}$.

Summary of Results:

1. $(f \circ f)(x) = 8^{8^x}$
2. $(f \circ g)(x) = 8^{x^8}$
3. $(g \circ f)(x) = 8^{8x}$
4. $(g \circ g)(x) = x^{64}$

Would you like further details or clarifications on any of these steps?

Here are some related questions to consider:

1. How do the rules for function composition work in general?
2. What are the differences between composing functions and multiplying them?
3. How does the composition of exponential functions behave compared to other types of functions?
4. What are the domain restrictions for these compositions?
5. How would the compositions change if $f(x)$ and $g(x)$ had different base forms?

Tip: Always evaluate the inner function first when performing function composition.